2022/062) Prove that every number in the sequence 16,1156,111556,⋯ is a perfect square.

We have for $n^{th}$ number it is n 1s followed by n-1 5's and ending with 6

of it is n 1's follwed by n 5's and add 1

n '1s $\frac{1}{9}(10^n-1)$

n 5's is   $\frac{5}{9}(10^n-1)$

so the number is  $\frac{1}{9}(10^n-1) * 10^n + \frac{5}{9}(10^n-1) + 1$

$= \frac{1}{9}(10^{2n}-10^n +  5* (10^n-1) + 9)$

$= \frac{1}{9}(10^{2n} +  4* 10^n + 4)$

$=\frac{1}{9} (10^n + 2)^2$

$=(\frac{1}{3}(10^n+2))^2$

which is  peffect square