To Prove 10^2n-1 + 1 is divisible by 11.

10^2n-1 + 1 is divisible by 11
Let P(n) = 10^2n-1 + 1
P(1) divisible by 11
Let us assume P(k) = 10^2n-1 + 1 is divisible by 11
P(k) = 10^2k-1 + 1 = 11M …………………( A)
To Prove P(k+1) is divisible by 11 using the result of A
P(k+1)² = 10^2(k+1)-1 + 1
            = 10^2k+2-1 + 1
            = 10^2k-1 . 10² + 1
            = (11M – 1) 10² + 1
            = (10²)11M – 100 + 1
            = (100)11M – 99
            = 11(100M – 9)
Which is divisible by 11.
hence the result.
P(K+1) is true.
By the Principle of mathematical induction, P(n) is true for all values of n where n N
Hence proved.