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

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

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