2009 AIME I Problems/Problem 7

Problem

The sequence $(a_n)$ satisfies $a_1 = 1$ and $5^{(a_{n + 1} - a_n)} - 1 = \frac {1}{n + \frac {2}{3}}$ for $n \geq 1$. Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$.

Solution

The best way to solve this problem is to get the iterated part out of the exponent: $$5^{(a_{n + 1} - a_n)} = \frac {1}{n + \frac {2}{3}} + 1$$ $$5^{(a_{n + 1} - a_n)} = \frac {n + \frac {5}{3}}{n + \frac {2}{3}}$$ $$5^{(a_{n + 1} - a_n)} = \frac {3n + 5}{3n + 2}$$ $$a_{n + 1} - a_n = \log_5{\left(\frac {3n + 5}{3n + 2}\right)}$$ $$a_{n + 1} - a_n = \log_5{(3n + 5)} - \log_5{(3n + 2)}$$ Since $a_1 = 1 = \log_5{5} = \log_5{(3(1) + 2)}$, we can easily use induction to show that $a_n = \log_5{(3n + 2)}$. So now we only need to find the next value of $n$ that makes $\log_5{(3n + 2)}$ an integer. This means that $3n + 2$ must be a power of $5$. We test $25$: $$3n + 2 = 25$$ $$3n = 23$$ This has no integral solutions, so we try $125$: $$3n + 2 = 125$$ $$3n = 123$$ $$n = \boxed{041}$$

See also

 2009 AIME I (Problems • Answer Key • Resources) Preceded byProblem 6 Followed byProblem 8 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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