Question

${10}^{(2n-1)}+1$ is divisible by

• A. 7
• B. 9
• C. 11
• D. 13

Answer 1

The correct option is C

11

Put n=1
${10}^{(2n-1)}+1={10}^{(2.1-1)}+1=11$
$P\left(n\right)={10}^{(2n-1)}+1$ is divisible by 11
$P\left(1\right)=11$ is divisible by 11.
Assume $P\left(k\right)$ is true
$\Rightarrow {10}^{(2k-1)}+1=11q$
To prove $P(k+1)$ is true.
${10}^{2(k+1)-1}+1={10}^{(2k+2-1)}+1$
$P(k+1)={10}^{2k+1}+1$
$={10}^2\left({10}^{2k-1}\right)+1$
$={10}^2({10}^{2k-1}+1-1)+1$
$P(k+1)={10}^2.11q-100+1$
$=11({10}^{2}q-9)$
$=11r$
Divisible by 11.
$\therefore P\left(n\right)$ is true $\forall n\in N$