2008 AIME II Problems/Problem 6

Problem

The sequence $\{a_n\}$ is defined by $a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.$ The sequence $\{b_n\}$ is defined by $b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.$ Find $\frac {b_{32}}{a_{32}}$ .

Solution

Rearranging the definitions, we have $\frac{a_n}{a_{n-1}} = \frac{a_{n-1}}{a_{n-2}} + 1,\quad \frac{b_n}{b_{n-1}} = \frac{b_{n-1}}{b_{n-2}} + 1$ from which it follows that $\frac{a_n}{a_{n-1}} = 1+ \frac{a_{n-1}}{a_{n-2}} = \cdots = (n-1) + \frac{a_{1}}{a_0} = n$ and $\frac{b_n}{b_{n-1}} = (n-1) + \frac{b_{1}}{b_0} = n+2$ . These recursions, $a_{n} = na_{n-1}$ and $b_{n} = (n+2)b_{n-1}$ , respectively, correspond to the explicit functions $a_n = n!$ and $b_n = \frac{(n+2)!}{2}$ (after applying our initial conditions). It follows that $\frac{b_{32}}{a_{32}} = \frac{\frac{34!}{2}}{32!} = \frac{34 \cdot 33}{2} = \boxed{561}$ .

Solution 2

It's very easy just to write out the first couple terms of each and find a pattern. The terms of ${a_n}$ are 1,1,2,6,and 24. The terms of ${b_n}$ are 1,3,12,60,and 360. When we divide ${b_n}$ by ${a_n}$ , we get 1,3,6,10, and 15, the triangular numbers! But since 1 corresponds to ${a_0}/{b_0}$ , the answer is the 33rd triangular number, which is 561.

2008 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2008 AIME II Problems/Problem 6

Contents

Problem

Solution

Solution 2

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