Answer :

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}. 10

^{2}+ 1

= (11M – 1) 10

^{2}+ 1

= (10

^{2})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.

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