The sum of first n terms of the series 1^2+2.2^2+3^2+2.4^2+5^2+5.6^2+…is (n(n+1)^2)/(2) when n i

step 1 Here we use M instead of n to denote this even number. So the sum of the first M terms of this s

The sum of first n terms of the series...

The sum of first $n$ terms of the series
$1^{2}+2.2^{2}+3^{2}+2.4^{2}+5^{2}+5.6^{2}+\ldots$ is $\frac{n(n+1)^{2}}{2}$
when $n$ is even. When $n$ is odd, the sum is
(A) $\frac{n^{2}(n+1)}{2}$
(B) $\frac{n(n+1)^{2}}{2}$
(C) $\left[\frac{n(n+1)}{2}\right]^{2}$
(D) $\frac{n(n+1)}{2}$

Key Concepts

Series Summation

Series summation involves determining a closed form or a simplified expression for the sum of the terms in a sequence. This often requires recognizing the underlying pattern or structure of the sequence and applying appropriate algebraic techniques or known summation formulas to simplify the process.

Pattern Recognition in Sequences

Pattern recognition is essential in handling series where terms may follow alternating or varied structures. By identifying how the sequence is built up—such as noticing alternating coefficients or differing behaviors in consecutive terms—one can split or transform the series in a way that makes the summation more manageable.

Parity-based Analysis

Parity-based analysis is the method of distinguishing between different cases based on whether an index or number of terms is even or odd. This technique is useful when the rule for generating the terms of a series changes with the parity of the index, leading to separate expressions or formulas for even and odd cases.

Algebraic Manipulation and Closed Form Derivation

Deriving a closed form expression for a series often involves algebraic manipulation, including factoring, expanding, and rearranging terms. This process transforms a potentially complex summation into an expression that directly computes the sum for any given number of terms without the need for iterative addition.

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