1999 AHSME Problems/Problem 20

Problem

The sequence $a_{1},a_{2},a_{3},\ldots$ statisfies $a_{1} = 19,a_{9} = 99$ , and, for all $n\geq 3$ , $a_{n}$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2$ .

Solution

Let $m$ be the arithmetic mean of $a_1$ and $a_2$ . We can then write $a_1=m-x$ and $a_2=m+x$ for some $x$ .

By definition, $a_3=m$ .

Next, $a_4$ is the mean of $m-x$ , $m+x$ and $m$ , which is again $m$ .

Realizing this, one can easily prove by induction that $\forall n\geq 3;~ a_n=m$ .

It follows that $m=a_9=99$ . From $19=a_1=m-x$ we get that $x=80$ . And thus $a_2 = m+x = \boxed{179}$ .

1999 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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1999 AHSME Problems/Problem 20

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