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. 

Answered by  | 4th Jun, 2014, 03:23: PM