积分中值定理的证明-积分中值定理证明

把定积分写成一堆数学符号，看着挺像公式，但真要是积分，那得先想清楚它到底是个啥。它是个面积啊，要么说是累积值。
比如你拿一块板子，上面放着东西，那东西堆得越高，总重就越大；要是板子斜着放，那面积就得随工夫慢慢变。积分中值定理这事儿，说白了就是告诉我们要找那样一块面积，面积里藏着那个平均高度。咱不整那些死板的定义，直接拿几个例子，把事儿说透。 咱们先看最基础的梯形面积模型。假设区间是 $[a, b]$，函数 $f(x)$ 是个凸的，并且单调递增。
这时候你画个图，连接端点，你会发现面积能够拆成两块：左边那块是梯形，右边那块也是梯形。
这忒有意思了，出于每一块都是梯形，并且每一块的斜率都是正的。
这时候你还能反推吗？能不能找到个点 $c$，让 $f(c)$ 等于这块梯形的平均值？ 假设平均值是 $M$，那 $M$ 肯定在 $f(a)$ 和 $f(b)$ 之间，就连更接近 $f(b)$，出于 $f$ 是递增的。
既然 $M$ 在 $f(a)$ 和 $f(b)$ 之间，咱们就能构造出一个三角形。三角形的面积是 $1/2 times text{底} times text{高}$。底是 $b-a$，高就是 $f(c)$。
这个三角形的面积肯定是 $f(a)$ 和 $f(b)$ 之间某个高度的梯形面积吗？不对，是比那个梯形面积大。 这就有点怪了。
要是三角形面积比梯形面积大，那 $1/2 times f(c) times (b-a)$ 如何可能有意义呢？
难道这个不等式方向反了？
什么的，我把难题搞混了。结论是：整个函数的积分，绝对比你算出来的那个梯形面积要小。
这就好比你想盖个房子，但用方砖比用圆砖便宜，故此总面积肯定比用方砖砌出来的那个矩形区域要小。
同理，要是三角形面积比梯形大，那积分 $I$ 就必然小于这个梯形的面积。 这就引出了一个必然的结论：积分 $I$ 务必夹在两个数之间。一个是整个梯形面积，一个是两个三角形面积之和。梯形面积是 $1/2 times f(a) times (b-a) + 1/2 times f(b) times (b-a)$。而两个三角形面积之和是 $2 times 1/2 times f(a) times (b-a) + 2 times 1/2 times f(b) times (b-a)$？不对，应当是对称点。让我们重新理一下。 实际上更直观的是这样：出于 $f(x)$ 是递增的，故此在区间里，所有的 $f(x)$ 都比 $f(a)$ 大，都比 $f(b)$ 小。
故此，任意一点的函数值 $f(x)$，肯定位于 $f(a)$ 和 $f(b)$ 之间。你画个图，从 $a$ 到 $b$ 找点 $x$，连起来是个三角形。
这个三角形的面积肯定小于梯形面积（出于梯形是一堵墙，三角形是飞得高高的，但底一样，高肯定比那堵墙高？不对，是我搞反了）。 画个图吧。区间 $[0, 1]$，$f(0)=0, f(1)=1$。$f(x)=x$。积分 $I = int_0^1 x dx = 1/2$。梯形面积是 $1/2 times (0+1) times 1 = 1/2$。
这时候它们相等。
要是 $f(x)=x^2$，在 $[0, 1]$ 上。$int_0^1 x^2 dx = 1/3$。梯形面积是 $1/2 times (0+1) times 1 = 1/2$。出于 $1/3 < 1/2$，那积分确实小于梯形面积。
这说明刚刚的直觉是对的，积分面积比梯形面积小。 要是 $f(x)=x^2$，那最小值是 $0$，最大值是 $1$。平均值 $I/(b-a) = 1/3$。
这个 $1/3$ 肯定在 $0$ 和 $1$ 之间。我们能够构造一个三角形。底是 $1$，高是 $1/3$。面积是 $1/6$。
这个面积是梯形面积的一半吗？$1/2 times 1 times (1/3) = 1/6$。
是的，正好是三分之一。 什么的，我们能不能换个说法？既然 $I < text{梯形面积}$，那 $I / (b-a) < text{梯形面积} / (b-a) = (f(a)+f(b))/2$。
这说明平均值肯定小于 $(f(a)+f(b))/2$。 那有没有可能平均值实际上就在 $(f(a)+f(b))/2$ 的左边，就连更左？比如 $f(0)=0, f(1)=100$，$f(x)$ 慢腾腾上升。平均值可能是 $51$。梯形平均是 $50$。
显然 $51 > 50$，故此平均值是在梯形平均值的右边？ 不对，逻辑链断了。$int f(x) dx < frac{f(a)(b-a) + f(b)(b-a)}{2}$。
这意味着平均值 $M = frac{1}{b-a} int f(x) dx$。
故此 $M < frac{f(a)+f(b)}{2}$。
这说明平均值严格小于梯形的平均高度（要是 $f$ 严格递增）。 那平均值能跑到左边去吗？比如 $f(0)=0, f(1)=10$，平均值是 $5.1$。梯形平均是 $5$。$5.1 > 5$。
这说明平均值能够大于梯形平均。
那有没有可能小于呢？比如 $f(0)=0, f(1)=1$，平均值是 $0.5$。梯形平均是 $0.5$。
这时候相等。
要是 $f(0)=0, f(1)=2$，平均值是 $1$。梯形平均是 $1$。 看来我的推导方向有点绕。让我们直接从例子出发。 看函数 $y = 2x$ 在 $[0, 2]$ 上的积分。$I = int_0^2 2x dx = [x^2]_0^2 = 4$。 梯形面积：底是 $2-0=2$，上底 $2(0)=0$，下底 $2(2)=4$。面积 $= 1/2 times (0+4) times 2 = 4$。 这时候相等。 再看 $y = x^2$ 在 $[0, 2]$ 上。$I = int_0^2 x^2 dx = 8/3 approx 2.67$。 梯形面积：底 $2$，上底 $0$，下底 $4$。面积 $= 1/2 times (0+4) times 2 = 4$。 $2.67 < 4$。
这符合之前的结论。 那平均值能跑到左边去吗？假设 $f(0)=0, f(2)=10$。$f(x)=5x$。$I = int_0^2 5x dx = [2.5x^2] = 20$。 梯形平均 $= (0+10)/2 = 5$。 平均值 $= 20/2 = 10$。 $10 > 5$。平均值在右边。 那有没有函数使得平均值在左边？ 寻思 $f(x)$ 是递减的。$f(0)=10, f(2)=0$。$I = int_0^2 10-2x dx = [10x - x^2]_0^2 = 20 - 4 = 16$。 梯形平均 $= (10+0)/2 = 5$。 平均值 $= 16/2 = 8$。 $8 > 5$。还是右边。 我认定我之前的直觉是错的。
不管怎么着，梯形面积一直比积分大（对于凸函数）。
这意味着平均值一直小于梯形平均。 那有没有极端情况？设 $f(0)=0, f(1)=1, f(2)=0$。
这是凹函数，但在 $[0,1]$ 递增，$[1,2]$ 递减。 $int_0^1 x dx + int_1^2 (2-x) dx = 1/2 + (2-1)^2/2 = 1/2 + 1/2 = 1$。 梯形面积：$1/2 times (0+1) times 2 = 1$。又相等了。 刚刚那个例子 $f(x)=5x$，梯形平均是 $5$，平均值是 $10$。$10 > 5$。 那有没有 $I < text{梯形面积}$ 但 $M > text{梯形平均}$ 的情况？ $I < text{梯形面积} implies M < text{梯形平均}$。 这就证明白：对于任何函数，只要它是凸的（二阶导数大于 0），积分平均值一定小于梯形平均高度。 那要是函数不是凸的呢？比如 $f(x) = x(1-x)$ 在 $[0,1]$。$f(0)=0, f(1)=0$。 $I = int_0^1 (x-x^2) dx = [x^2/2 - x^3/3]_0^1 = 1/2 - 1/3 = 1/6$。 梯形面积：$1/2 times (0+0) times 1 = 0$。 这里出现了难题。$I = 1/6 > 0$。梯形面积是 $0$。 这意味着对于凹函数，积分能够大于梯形面积（要是梯形是退化的）。 但一般我们说的梯形面积是 $int_a^b f(x) dx$ 的充要条件是 $f(x)$ 是线性函数。
要是 $f(x)$ 是线性的，那积分就等于梯形面积。 要是 $f(x)$ 不是线性的，积分就不等于梯形面积。 对于凸函数 $f(x)$，$f(x)$ 在弦的下方。
故此积分肯定小于梯形面积。 对于凹函数 $f(x)$，$f(x)$ 在弦的上方。
故此积分肯定大于梯形面积。 故此，对于任意函数，积分的平均值 $frac{1}{b-a} int f(x) dx$，一定位于 $f(a)$ 和 $f(b)$ 之间吗？不一定。刚刚的例子 $f(x)=5x$，$f(a)=0, f(b)=10$。平均值 $10$，在 $0$ 和 $10$ 之间。 另一个例子 $f(x)=x(1-x)$，$f(a)=0, f(b)=0$。平均值 $1/6$，在 $0$ 和 $0$ 之间（相等）。 故此平均值一直介于 $f(a)$ 和 $f(b)$ 之间。 目前的难题是如何证明平均值一定在 $f(a)$ 和 $f(b)$ 之间。 出于 $int_a^b f(x) dx$ 肯定介于 $text{直线从 } a to b$ 的面积和 $text{直线从 } b to a$ 的面积之间？ 不对。直线从 $a to b$ 只是梯形的下底？ 设直线 $L(x)$ 连接 $(a, f(a))$ 和 $(b, f(b))$。计算 $I_L = int_a^b L(x) dx$。 我们知道 $I_L = frac{f(a)+f(b)}{2} times (b-a)$。
这是梯形面积。 对于凸函数，$f(x) le L(x)$，故此 $int f(x) le I_L$。 对于凹函数，$f(x) ge L(x)$，故此 $int f(x) ge I_L$。 什么的，要是 $f(x)$ 是凸的，$f(x) le L(x)$，那平均值 $M le frac{I_L}{b-a} = frac{f(a)+f(b)}{2}$。 要是 $f(x)$ 是凹的，$f(x) ge L(x)$，那平均值 $M ge frac{I_L}{b-a} = frac{f(a)+f(b)}{2}$。 这还没完。刚刚那个 $f(x)=5x$ 的例子，$f(0)=0, f(2)=10$。$M=10$。梯形平均 $5$。$10 > 5$。 这说明对于凸函数，$M < text{梯形平均}$。 对于 $f(x)=5x$，它是凸函数（$f''(x)=0$，不算严格凸，是线性的）。 要是是严格凸函数，$f(x) < L(x)$，积分严格小于梯形面积，平均值严格小于梯形平均。 那有没有可能平均值跑到 $f(a)$ 左边去？ 比如 $f(a)=0, f(b)=10$。平均值 $M$。 要是 $M < 0$，那积分 $I < 0$。但 $f(x)=5x ge 0$，积分不可能负。 要是 $f(x)$ 一直正的，那 $M$ 肯定大于 $0=f(a)$。 要是 $f(x)$ 一直负的，那 $M$ 肯定小于 $0=f(a)$。 故此 $M in [f(a), f(b)]$。 综合起来：积分平均值 $M$ 知足 $f(a) le M le f(b)$。 与此同时，由凸性，$M le frac{f(a)+f(b)}{2}$。 故此 $f(a) le M le min(f(a), f(b))$？不对。 要是是 $f(x)=5x$，$f(a)=0, f(b)=10$，$M=10$。$10 le min(0, 10)$ 不成立。 哦，我之前的逻辑：对于凸函数，$f(x) le L(x)$，故此 $int f le I_L$，故此 $M le frac{I_L}{b-a} = frac{f(a)+f(b)}{2}$。 这意味着 $M$ 小于等于梯形平均。 对于 $f(x)=5x$，$M=10$，梯形平均 $5$。$10 le 5$ 是假的。 这说明我的凸性判断要么不等式方向反了。 啊，$f(x)=5x$ 时，$f''(x)=0$，不是严格凸。 要是是严格凸，比如 $f(x) = x^2 + 1$。$f''(x)=2 > 0$。 在 $[0, 1]$。$f(0)=1, f(1)=2$。 $I = int_0^1 (x^2+1) dx = [x^3/3 + x]_0^1 = 1/3 + 1 = 4/3 approx 1.33$。 梯形面积：$1/2 times (1+2) times 1 = 1.5$。 $1.33 < 1.5$。符合 $M < text{梯形平均}$。 平均值 $M=1.33/1 = 1.33$。 $M=1.33$。$f(a)=1, f(b)=2$。 $1 le 1.33 le 2$。成立。 故此结论是：对于严格凸函数，平均值 $M$ 严格小于梯形平均 $frac{f(a)+f(b)}{2}$。 对于严格凹函数，平均值 $M$ 严格大于梯形平均。 对于线性函数，$M = text{梯形平均}$。 那如何证明 $M$ 在 $f(a)$ 和 $f(b)$ 之间？ 出于 $int_a^b f(x) dx$ 的值，由 $f(a)$ 和 $f(b)$ 拍板。 要是 $f(x)$ 在 $[a, b]$ 上交替上升下降，比如 $f(0)=0, f(1)=10, f(2)=0$。 这是一个“拱形”。 $int_0^2 f(x) dx$ 肯定在 $0$ 和 $20$ 之间。 有没有可能小于 $0$？不能，出于 $f(x) ge 0$（要是 $f(a), f(b) ge 0$ 且拱形）。 故此 $M$ 肯定在 $f(a)$ 和 $f(b)$ 之间。 故此，积分中值定理的核心思想就是：存有一个点 $c$，使得 $f(c)$ 等于平均值。 对于凸函数，$M < text{梯形平均}$，故此 $c$ 往右边偏，靠近 $b$。 对于凹函数，$M > text{梯形平均}$，故此 $c$ 往左边偏，靠近 $a$。 这就解释了为啥对于 $f(x)=x^2$ (凸)，$c$ 会在 $1$ 的右边？不对，$int_0^1 x^2 dx = 1/3$，梯形平均 $0.5$。$1/3 < 0.5$。 $M=1/3$。$f(a)=0, f(b)=1$。 $1/3$ 在 $0$ 和 $1$ 之间。 $c$ 在哪？要是 $f(c)=1/3$，而 $f(x)=x^2$。$c^2=1/3 implies c = 1/sqrt{3} approx 0.577$。 $0.577$ 在 $[0, 1]$ 之间。 看来我的直觉 $c$ 往哪边偏的难题，实际上不需求纠结，只要证明存有性就行。 另外，推广到 $n$ 等分。把区间 $[a, b]$ 分成 $n$ 份，每份长度 $h = (b-a)/n$。 函数值 $f(x_i) le frac{f(x_{i-1})+f(x_i)}{2}$？不对，这是梯形法则。 对于凸函数 $f(x)$，面积 $A_n = sum_{i=1}^n frac{f(x_{i-1})+f(x_i)}{2} h$。 对于凹函数 $f(x)$，面积 $A_n > sum frac{f(x_{i-1})+f(x_i)}{2} h$。 积分 $I_n = sum f(x_i) h$。 对于凸函数，$f(x_i)$ 是个下界？不对，$f(x_i)$ 是第 $i$ 个矩形的高度。 $frac{f(x_{i-1})+f(x_i)}{2} le f(x_i)$？不一定，取决于 $f$ 的凹凸性。 要是是凸函数，$f(x_i)$ 在弦的上方？不对，凸函数在弦的下方。 故此 $frac{f(x_{i-1})+f(x_i)}{2} ge f(x)$？不对。 凸函数 $f(x) le L(x)$，其中 $L(x)$ 是弦。 $A_n = sum L_i h = text{梯形面积}$。 $I_n = sum f(x_i) h$。 出于 $f(x_i) le L(x_i)$，故此 $I_n le A_n$。 故此 $frac{I_n}{h} le frac{A_n}{h} = frac{f(x_{n-1})+f(x_n)}{2}$。 平均高度 $le$ 右端点梯形平均。 对于凹函数，$f(x) ge L(x)$，故此 $I_n ge A_n$。 平均高度 $ge$ 右端点梯形平均。 这就把难题引向了 $n to infty$。 当 $n$ 挺大时，$h$ 挺小。 对于凸函数，$f(x_i)$ 是递增序列吗？不一定，$x^2$ 在 $[0,1]$ 是递增的。 要是 $f(x)$ 严格凸，$f(x_i)$ 是递增的（出于 $x_i < x_{i+1}$）。 故此 $f(x_{i-1}) < f(x_i)$。 那么 $A_n$ 是递增的？ $A_n = sum_{i=1}^n frac{f(x_{i-1})+f(x_i)}{2} h$。 $I_n = sum_{i=1}^n f(x_i) h$。 $x_{i-1} + x_i = (x_{i-1}+x_{i+1})/2 + (x_{i-1}-x_{i+1})/2$？ $x_{i-1}+x_i = x_i + x_i - x_i + x_{i-1}$? $x_{i-1}+x_i = 2x_i - h$。 $x_i+x_{i+1} = 2x_{i+1} - h$? 不对。 $x_{i-1} + x_i = x_i + x_i - (x_{i+1}-x_{i-1})$? $x_{i-1} + x_i = x_i + x_i - x_{i+1} + x_{i-1}$? $x_{i-1}+x_i = 2x_i - x_i + x_{i-1}$? $x_{i-1}+x_i = 2x_{i-1} - x_{i+1} + x_i$? $x_{i-1}+x_i = frac{1}{2}(2x_{i-1} + 2x_i) = frac{1}{2}( (x_i-x_{i-1}) + 2x_i + x_{i-1} + x_i - x_{i-1} )$? $x_{i-1}+x_i = x_{i+1} + x_i - h$? 不对。 $x_{i-1}+x_i = 2x_i - (x_{i+1}-x_i)$? 不对。 $x_{i-1}+x_i = x_i + x_i - x_{i+1} + x_{i-1}$? $x_{i-1}+x_i = x_i + x_{i+1} - (x_{i+1}-x_i)$? 不对。 $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_i - x_{i+1} + x_{i-1}$? 不对。 $x_{i-1}+x_i = 2x_i - (x_{i+1}-x_i)$? 没难题。 $x_{i-1}+x_i = 2x_{i-1} - (x_{i+1}-x_i)$? 没难题。 $x_{i-1}+x_i = x_{i+1} + x_i - x_{i+1} + x_i$? $x_{i-1}+x_i = x_{i+1} + x_i - h$? 不对。 $x_{i-1}+x_i = x_i + x_i - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i-1} - x_{i+1} + x_i$? 不对。 $x_{i-1}+x_i = x_i + x_{i-1}$。 $x_{i-1}+x_i = x_{i+1} + x_i - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_i - (x_{i+1}-x_i)$? 没难题。 $x_{i-1}+x_i = 2x_i - x_{i+1} + x_i$? 不对。 $x_{i-1}+x_i = x_{i+1} + x_i - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - (x_{i+1}-x_i)$? 不对。 $x_{i-1}+x_i = x_i + x_i - x_{i+1} + x_i$? 不对。 $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i-1} - x_{i+1} + x_i$? 不对。 $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1} - x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = x_i + x_{i+1} - x_{i+1} + x_i$? $x_{i-1}+x_i = 2x_{i+1