102n - 1 + is divisible by 11 – InterviewSolution

1.	102n - 1 + is divisible by 11
Answer» Let P(n): 10^2n-1 +1, is divisible by 11. For n = 1, P(1):10^2-1+1, is divisible by 11, which is true. P(1) is true. Let us assume P(k) is true for some k∈N. i.e., 10^2n-1+1 is divisible by 11. ⇒10^2-1 +1 = 1 W, d∈N. Consider 10^2(k+1)-1+1 = 10^2k+1+1 = 10^2k-1 10²+ 1 = (11d - 1)10²+1 = 11d(10²) -100 + 1 = 11(100d) - 99 = 11(100d - 9) ∴ 10^{2(k + 1)-1} +1 is divisible by 11. Thus, P(k) ⇒ P(k +1) Hence, by mathematical induction, P(n) is true for all n∈N

102n - 1 + is divisible by 11

Answer»

Let P(n): 10^2n-1 +1, is divisible by 11.

For n = 1, P(1):10^2-1+1, is divisible by 11, which is true.

P(1) is true. Let us assume P(k) is true for some k∈N.

i.e., 10^2n-1+1 is divisible by 11.

⇒10^2-1 +1 = 1 W, d∈N.

Consider 10^2(k+1)-1+1 = 10^2k+1+1

= 10^2k-1 10²+ 1 = (11d - 1)10²+1

= 11d(10²) -100 + 1 = 11(100d) - 99

= 11(100d - 9)

∴ 10^{2(k + 1)-1} +1 is divisible by 11.

Thus, P(k) ⇒ P(k +1)

Hence, by mathematical induction, P(n) is true for all n∈N