If n is a positive integer, then 2 4^ 2 n+1 +3^ 3 n+1 is divisible by

Step-by-step solution

Given expression, p(n)=2 4^ 2 n+1 +3^ 3 n+1 Let n=1p(1)=2 4²⁺¹+3³⁺¹=209 which is divisible by 11 . Let p(k) is also divisible by 11 . p(k)=2 4^ 2 k+1 +3^ 3 k+1 is divisibly by 11 ...(i) Now we will prove that p(k+1) is true. p(k+1)=2 4^ 2(k+1)+1 +3^ 3(k+1)+1 aligned & =2 4^ 2 k+1 16+3^ 3 k+1 27 & =2 4^ 2 k+1 16+3^ 3 k+1 16+3^ 3 k+1 11 & =16 (2 4^ 2 k+1 +3^ 3 k+1 )+3^ 3 k+1 11 aligned Divisible by 11 from Eq. (i) p(k+1) is also true. p(n) is divisible by 11 .