Sum to n terms of the series 2+5+14+41+…is (A) (n)/(2)+(1)/(4)(3^n-1) (B) (n)/(2)+(3)/(4)(3^n-1)

step 1 In this question, the series is 2 plus 2 plus 3, 2 plus 2 plus 3 and so on. The difference of co

Sum to n terms of the series 2+5+14+41+…is (A)...

Sum to $n$ terms of the series $2+5+14+41+\ldots$ is
(A) $\frac{n}{2}+\frac{1}{4}\left(3^{n}-1\right)$
(B) $\frac{n}{2}+\frac{3}{4}\left(3^{n}-1\right)$
(C) $\frac{n}{2}+\frac{1}{2}\left(3^{n}-1\right)$
(D) None of these

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Key Concepts

Series Summation Techniques

These techniques involve expressing the general term of a series and applying known formulas to sum the series over a finite number of terms. Understanding how to decompose a series into simpler parts, such as a constant part and a geometric part, is a key strategy in solving summation problems.

Geometric Series

A geometric series is a sequence in which each term is a constant multiple (the common ratio) of the previous term. Recognizing a geometric progression within a set of differences or the sequence itself allows the use of the geometric series sum formula, which is vital for finding closed-form expressions.

Finite Differences

Finite differences help in discovering patterns in the increments between consecutive terms. When these differences form a recognizable sequence (like a constant or a geometric sequence), it offers insight into the formula generating the original sequence.

Sequence Analysis

This involves identifying the pattern or rule that defines the progression of terms in a sequence. In many problems, carefully analyzing how each term relates to its predecessors can reveal underlying structures, such as arithmetic or geometric patterns, or even a combination of both.

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