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

Explanation

Let `P (n) =10^(2n-1) +1`
For n =1
`P(1)=10^(2xx1-1) +1 =10 +1 =11=11xx1`
Which is divisible by 11.
`:. `P(n) is true fon n=1
Let P (n) be true for n=K
`rArr {10^(2k-1)+1]` is divisible by 11
`:. 10^(2k-1) +1 =11 lambda , " where " lambda in I`
Now `P(k+1) =10^(2(K+1)-1)=1`
`rArr P(k+1) =10^(2k-1) . 10^(2) +1`
`=10^(2k-1).10^(2)+10^(2)-10^(2)+1`
`=10^(2)"("10^(2k-1)+1")"-100 +1`
`=100 . 11 lambda -99 =11 (100lambda-9)`
[From equation (1)]
`rArr P (K+1)` is divisible by 11
`rArr P (n) ` is also true for n=K+1
Hence by the principle of mathematical induction P(n) is true for all natural numbers n.