The sum of the first n terms of the series I^2+2 ·2^2 +3^2+2 ·4^2+5^2+2 ·6^2+…is (n(n+1)^2)/(2)

step 1 We assume m is an even number and hence 1 squared plus 2 times 2 squared plus 3 squared plus etc

The sum of the first n terms of the series I^2+2 ·2^2 +3^2+2...

The sum of the first $\mathrm{n}$ terms of the series $\mathrm{I}^{2}+2 \cdot 2^{2}$ $+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots$ is $\frac{n(n+1)^{2}}{2}$ when $n$ is even. When $n$ is odd the sum is $\quad$ [2004] (A) $\frac{3 n(n+1)}{2}$ (B) $\frac{n^{2}(n+1)}{2}$ (C) $\frac{n(n+1)^{2}}{4}$ (D) $\left[\frac{n(n+1)}{2}\right]^{2}$

The sum of the first $\mathrm{n}$ terms of the series $\mathrm{I}^{2}+2 \cdot 2^{2}$ $+3^{2}+2 \cdot 4^{2}+5^{2}+2 \cdot 6^{2}+\ldots$ is $\frac{n(n+1)^{2}}{2}$ when $n$ is
even. When $n$ is odd the sum is $\quad$ [2004]
(A) $\frac{3 n(n+1)}{2}$
(B) $\frac{n^{2}(n+1)}{2}$
(C) $\frac{n(n+1)^{2}}{4}$
(D) $\left[\frac{n(n+1)}{2}\right]^{2}$

Key Concepts

Pattern Recognition in Series

This key concept involves identifying recurring structures or rules in the sequence of terms. By observing how each term is generated, such as recognizing alternations or multipliers applied to specific positions in the series, one can group terms or perform strategic substitutions that simplify the process of summing the series.

Closed-form Expression

A closed-form expression is a finite expression that allows the computation of the sum of a series without requiring iterative addition. It consolidates the sequence's rule into a single formula that can be applied for any number of terms, providing an exact answer efficiently and highlighting the underlying structure of the series.

Finite Series Summation

This concept involves techniques for evaluating the sum of a sequence of numbers defined by a specific formula. It often requires using known formulas for sums of powers of integers, applying algebraic manipulations, and sometimes separating the series into parts that are easier to sum. This process leads to a closed?form expression that directly computes the sum for any given number of terms.

Parity Analysis

Parity analysis is the method of differentiating between even and odd indices in a sequence. In many series problems, the behavior of the sequence changes based on whether the term number is even or odd, leading to distinct expressions or adjustments in the summation formula. Recognizing and handling these differences is crucial for correctly deriving formulas for each case.

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