Question

Find the sum of the series:

5 + 7 + 13 + 31 + 85 + …. To n terms

Answer 1

In the given question we need to find the sum of the series.

For that, first, we need to find the n^th term of the series so that we can use summation of the series with standard identities and get the required sum.

The series given is 5 + 7 + 13 + 31 + 85 + …. + n terms.

This question can be solved by the method of difference.

Note:

Consider a sequence a₁, a₂, a₃ …such that the Sequence a₂ – a₁, a₃ – a₂… is either an. A.P. or a G.P.

The n^th term, of this sequence, is obtained as follows:

Since the terms within the brackets are either in an A.P. or a G.P, we can find the value of a_n, the n^th term.

Thus, we can find the sum of the n terms of the sequence as,

By using the method of difference, we can find the n^th term of the expression.

In the resulting series obtained, starting from 2, 6, 18…forms a GP.

So, the n^th term forms a GP, with the first term, a = 2; common ratio, r = 3.

The required n^th term of the series is the same as the sum of n terms of GP and 5.

Note:

So, for the given series, we need to find,

The first term in (a) is a GP, with the first term, a = 3 and common ratio, r = 3.