Answer : (C) = 7
For n = 1, 23n – 1 = 23 – 1 = 8 – 1 = 7, which is divisible by 7, and not divisible by any other alternative given.
∴ We shall prove 23n – 1 divisible by 7 for all n∈N.
Let T(n) = 23n – 1 is divisible by 7.
Basic Step:
For n = 1, T(1) = 23 – 1 = 8 – 1 = 7 is divisible by 7 is true.
Induction Step:
Assume T(k) to be true, i.e.,
T(k) = 23k – 1 is divisible by 7 ⇒ 23k – 1 = 7m, m∈N
⇒ 23k = 7m + 1 ...(i)
Now 23(k + 1) – 1 = 23k + 3 – 1 = 23 . 23k – 1 = 8.23k – 1 ...(From (i))
= 8. (7m + 1) – 1 = 56m + 7 = 7 (8m + 1)
⇒ 23(k + 1) – 1 is divisible by 7
∴ T(k + 1) is true whenever T(k) is true.
⇒ 23n – 1 is divisible by 7 for all n∈N.