Prove by the principle of induction that for all `n N ,\ (10^(2n-1)+1)` is divisible by 11.

Explanation

Here, `P(n) = 10^(2n-1)+1`

`P(1) = 10^1+1 =11`, which is divisible by 11.

Now, we assume, for any number ` k, P(k)` is divisible by 11.

Then,`P(k) = 10^(2k-1)+1` is divisible by 11. `->(1)`

Now, we have to prove `P(k+1)` is also divisible by 11.

`P(k+1) = 10^(2(k+1)-1)+1`

`=10^(2k+1)+1`

`=10^2*10^(2k-1)+100-99`

...