HMMT November

Number theory Difficulty 4.8 AIME Find the answer

Find the number of terms $n \leq 2012$ such that $a_{n}=\frac{3^{n+1}-1}{2}$ is divisible by 7.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

To demonstrate that $7 \mid a_{n}=\frac{3^{n+1}-1}{2}$ is equivalent to showing that $3^{n+1}-1 \equiv 0 \bmod 7$ . However $3^{n+1}-1=0 \bmod 7 \Longrightarrow n+1 \equiv 0 \bmod 6$ , hence the values of $n \leq 2012$ are $5,11,17, \ldots, 2009$ , which is in total $\frac{2009+1}{6}=335$ possible terms.

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