7

Part of 2009 AIME Problems

Problems(2)

Source: 2009 AIME I #7

(a_n)

satisfies a_1 \equal{} 1 and \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}} for

n \geq 1

k

be the least integer greater than

1

a_k

is an integer. Find

k

logarithmsinductioninvariantAMC

Odd/Even Factorial Division

Source: AIME 2009II Problem 7

n!!

to be n(n\minus{}2)(n\minus{}4)\ldots3\cdot1 for

n

odd and n(n\minus{}2)(n\minus{}4)\ldots4\cdot2 for

n

even. When \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!} is expressed as a fraction in lowest terms, its denominator is

2^ab

b

\displaystyle \frac{ab}{10}

factorialAMCAIMEfloor functionfunction