Sum to n terms of the series 1^3+3.2^3+3^3+3.4^3+5^3 +…., where n is even, is (A) (n^2(n^2-3 n+1

step 1 Hello everyone, here we are given that S -N sum of n terms is equal to 1 cube plus 3 into 2 cube

Sum to n terms of the series 1^3+3.2^3+3^3+3.4^3+5^3 +….,...

Sum to $n$ terms of the series $1^{3}+3.2^{3}+3^{3}+3.4^{3}+5^{3}$ $+\ldots .$, where $n$ is even, is
(A) $\frac{n^{2}\left(n^{2}-3 n+1\right)}{2}$
(B) $\frac{n^{2}\left(n^{2}+3 n+1\right)}{2}$
(C) $\frac{n^{2}\left(n^{2}+3 n+1\right)}{4}$
(D) None of these

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Key Concepts

Series Summation Techniques

Series summation techniques involve methods for determining the closed-form expression of a series. This concept includes identifying the general term of a sequence and applying strategies such as pairing terms, grouping, or using indices transformations to simplify the sum of a series.

Pattern Recognition in Sequences

Pattern recognition is critical in sequence analysis, as it enables one to determine the rule or relationship that generates the sequence. Once the pattern is identified, particularly when it involves alternating or structured modifications of terms (such as even and odd-indexed elements), it becomes possible to express the general term in a usable form for summation.

Sum of Cubes

The sum of cubes concept pertains to understanding and utilizing formulas related to the summation of cubic terms. This includes not only the standard formula for the sum of cubes of consecutive integers but also how to adapt or combine such formulas when the series includes modifications or additional coefficients.

Algebraic Manipulation

Algebraic manipulation encompasses the skills needed to simplify complex algebraic expressions. When summing series, it often involves expanding sums, factoring expressions, and combining like terms to derive a simplified, closed-form result.

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