Answer : In the given question we need to find the sum of the series.
For that, first, we need to find the nth 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 nth term, of this sequence, is obtained as follows: S = a1 + a2 + a3 +…+ an–1 + an→ (1)
S = a1 + a2 +…+ an–2 + an–1 + an → (2) Subtracting (2) from (1),
We get, an = a1+ [(a2–a1) + (a3–a2) +… (an – an–1)].
Since the terms within the brackets are either in an A.P. or a G.P, we can find the value of an, the nth term.
Thus, we can find the sum of the n terms of the sequence as,