The value of (n-2)^2+(n-4)^2+(n-6)^2+…to n terms is (A) (n)/(3)(n^2+2) (B) (n)/(2)(n^2+3) (C) (n

step 1 The formula for the S term of this sequence, AI, equals n minus 2i and then squared. So the sum

The value of (n-2)^2+(n-4)^2+(n-6)^2+…to n terms is (A)...

The value of $(n-2)^{2}+(n-4)^{2}+(n-6)^{2}+\ldots$ to $n$ terms is
(A) $\frac{n}{3}\left(n^{2}+2\right)$
(B) $\frac{n}{2}\left(n^{2}+3\right)$
(C) $\frac{n}{3}\left(n^{2}-2\right)$
(D) $\frac{n}{2}\left(n^{2}-3\right)$

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Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In general, if the first term is a and the common difference is d, the nth term can be expressed as a + (n–1)d. Recognizing an arithmetic sequence is crucial for identifying patterns and applying appropriate sum formulas in the context of series summation.

Sum of Squares

The sum of squares involves summing the squares of terms in a sequence. When dealing with an arithmetic sequence whose terms are squared, it is necessary to use well-known summation formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers to simplify the expression. This approach leads to a general solution that often yields a formula in terms of n.

Summation Formulas

Standard summation formulas such as the sum of the first n numbers (n(n+1)/2) and the sum of the squares of the first n numbers (n(n+1)(2n+1)/6) are fundamental tools for handling series summations. These formulas help in reducing complex series to a manageable algebraic expression, especially when the series is derived from an arithmetic progression whose terms are manipulated by squaring.

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