1+3+3²+.....+3^n-1={3^n-1-1}2

prove the following with the help of principle of mathematical induction

Let the given statement be P(n), i.e.,

P(n): 1 + 3 + 3² + …+ 3^n–1 =

For n = 1, we have

P(1): 1 =, which is true.

Let P(k) be true for some positive integer k, i.e.,

We shall now prove that P(k + 1) is true.

Consider

1 + 3 + 3² + … + 3^k–1 + 3^{(k+1) – 1}

= (1 + 3 + 3² +… + 3^k–1) + 3^k

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.