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in Mathematical Induction by (22.8k points)
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For all n∈N, (23n – 1) will be divisible by

(a) 25 

(b) 8 

(c) 7 

(d) 3

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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.

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