# Prove the following using the principle of mathematical induction for all n ∈ N102n – 1 + 1 is divisible by 11.

Let the given statement be P(n), as

is divisible by 11.

First, we check if it is true for n = 1,

It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):102k - 1 + 1 = 11m where m N

102k - 1 = 11m - 1 ………….(1)

We shall prove that P(k + 1)is true,

P(k + 1):102k + 1 + 1

102k - 1.102 + 1

(11m - 1).100 + 1 From equation(1)

1100m - 100 + 1

1100m - 99

11(100m - 9)

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n N.

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