23n 1 Is Divisible By 7 For All Natural Numbers

23n – 1 is divisible by 7, for all natural numbers .

Answers:

Let P(n): 2³ⁿ – 1 is divisible by 7
Now, P( 1): 2³ – 1 = 7, which is divisible by 7.
Hence, P(l) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k): 2^3k – 1 is divisible by 7.
or  2^3k -1 = 7m, m∈ N  (i)
Now, we have to prove that P(k + 1) is true.
P(k+ 1): 2³(k+1)– 1
= 2^3k.23 -1
= 8(7 m + 1) – 1
= 56 m + 7
= 7(8m + 1), which is divisible by 7.
Thus, P(k + 1) is true whenever P(k) is true.
So, by the principle of mathematical induction P(n) is true for all natural numbers n.