Question

Prove by mathematical induction 10 2n 1 1 is divisible by 11

Answer 1

Answer:

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n = 1

Answer 2

Prove that 102n – 1 + 1 is divisible by 11 for all n N.

Asked by Topperlearning User | 4th Jun, 2014, 01:23: PM

Expert Answer:

Let P(n): 102n – 1 + 1 is divisible by 11

P(1): 10 + 1 = 11which is divisible by 11.

Thus P(n) is true for n = 1.

Let P(k) be true for some natural number k.

i.e. 102k – 1 + 1 is divisible by 11.

Let 102k – 1 + 1 = 11d

Now we prove that P(k + 1) is true whenever P(k) is true.

Now, P(k+1): 102k+1 + 1 = (11d – 1)100 + 1 = 11 (100d – 9),which is divisible by 11.

P(k + 1) is true.

Thus P(k +1) is true whenever P(k) is true.

By principle mathematical induction 102n – 1 + 1 is divisible by 11 for all n N.