2008 Mock ARML 1 Problems/Problem 5

Problem

The positive real numbers $x_1, x_2, \ldots, x_{10}$ are in arithmetic progression in that order. They also satisfy

$x_1^2 - x_2^2 + x_3^2 - \cdots - x_{10}^2 = x_1 + x_2 + \cdots + x_{10}.$

Compute the common difference of this arithmetic progression.

Solution

Let $d$ be the common difference of the progression. Then, by difference of squares,

$\begin{align*} (x_1 - x_2)(x_1 + x_2) + (x_3 - x_4)(x_3 + x_4) + \cdots &= x_1 + x_2 + \cdots + x_{10}\\ -d(x_1 + x_2 + \cdots + x_{10}) &= x_1 + x_2 + \cdots + x_{10}\\ d &= \boxed{-1} \end{align*}$

2008 Mock ARML 1 (Problems, Source)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8

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2008 Mock ARML 1 Problems/Problem 5

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