Solve this:
Dear student
Let the given statement be P(n), i.e.,
P(n): 23n – 1 is divisible by 7.
It can be observed that P(n) is true for n = 1 since 23 × 1 – 1 = 8 – 1 = 7, which is divisible by 7.
Let P(k) be true for some positive integer k, i.e.,
P( k ): 23k – 1 is divisible by 7.
∴23k – 1 = 7m; where m ∈ N … (1)
We shall now prove that P(k + 1) is true whenever P(k) is true.
Consider
Therefore, 23n – 1 is divisible by 7.
Thus, P(k + 1) is true whenever P(k) is true.
Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.
Regards