IIT-JEE 1996Question 1
Using mathematical induction prove that for every integer $n\ge1$, $\left(3^{2^{n}}-1\right)$ is divisible by $2^{n+2}$ but not by $2^{n+3}$.
Mathematical Reasoning
Answerproved
Solution
$n=1$: $3^2-1=8=2^3$ exactly divisible by $2^3$ not $2^4$. If $3^{2^n}-1=2^{n+2}m$ with $m$ odd, then $3^{2^{n+1}}-1=(3^{2^n}-1)(3^{2^n}+1)=2^{n+2}m(2^{n+2}m+2)=2^{n+3}m(2^{n+1}m+1)$, with $m(2^{n+1}m+1)$ odd. So divisible by $2^{n+3}$ but not $2^{n+4}$.