If am be the m th term of an A.P., then a1^2-a2^2+a3^2-a4^2+….+a2 n-1^2-a2 n^2= (A) (n-1)/(2 n-1

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If am be the m th term of an A.P., then...

If $a_{m}$ be the $m$ th term of an A.P., then $a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+\ldots .+a_{2 n-1}^{2}-a_{2 n}^{2}=$ (A) $\frac{n-1}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)$ (B) $\frac{n}{2 n-1}\left(a_{2 n}^{2}-a_{1}^{2}\right)$ (C) $\frac{n}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)$ (D) None of these

If $a_{m}$ be the $m$ th term of an A.P., then $a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+\ldots .+a_{2 n-1}^{2}-a_{2 n}^{2}=$
(A) $\frac{n-1}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)$
(B) $\frac{n}{2 n-1}\left(a_{2 n}^{2}-a_{1}^{2}\right)$
(C) $\frac{n}{2 n-1}\left(a_{1}^{2}-a_{2 n}^{2}\right)$
(D) None of these

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Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence where each term is formed by adding a constant difference to the previous term. Understanding the general formula, a_m = a_1 + (m - 1)d, is essential as it allows one to express any term in the sequence in terms of the first term and the common difference, which is key to analyzing patterns and solving related problems.

Difference of Squares Identity

The difference of squares identity, which states that a^2 - b^2 = (a + b)(a - b), is a vital algebraic tool. It can be used to factor and simplify expressions, especially when dealing with squares of terms in sequences or series, as seen in problems that involve subtracting the squares of consecutive terms.

Alternating Series Summation

Alternating series involve terms that switch signs, often leading to cancellation effects. Recognizing the structure of an alternating series and pairing terms appropriately, sometimes by leveraging the difference of squares identity, is crucial in simplifying and calculating the sum of such series.

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