If the sum of n terms of an A.P. is cn (n-1), where c ≠0, then sum of the squares of these terms

step 1 We let TI denotes the S term of this arithmetic progression. And so the S term equals the sum of

If the sum of n terms of an A.P. is cn (n-1), where c ≠0,...

If the sum of $n$ terms of an A.P. is cn $(n-1)$, where $c \neq 0$, then sum of the squares of these terms is
(A) $c^{2} n^{2}(n+1)^{2}$
(B) $\frac{2}{3} c^{2} n(n-1)(2 n-1)$
(C) $\frac{2 c^{2}}{3} n(n+1)(2 n+1)$
(D) None of these

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence in which each term after the first is obtained by adding a constant difference to the previous term. This constant difference is known as the common difference. Understanding APs is fundamental when dealing with problems involving sequences and series, as many real-world phenomena and mathematical models use APs to represent linear relationships.

Sum of Terms of an AP

The sum of the first n terms of an AP is typically given by a formula that relates the first term, the common difference, and the number of terms. This formula, generally expressed as S? = n/2[2a + (n - 1)d], is crucial for finding relationships or constraints on the parameters of an AP when specific conditions on the sum are provided.

General Term of an AP

The nth term of an arithmetic progression is defined by the equation a? = a + (n - 1)d, where a is the first term and d is the common difference. This general term is essential for expressing any term in the sequence and is often used as a basis for further operations such as squaring terms to compute sums of squares.

Sum of Squares of an AP

Calculating the sum of the squares of the terms in an AP involves squaring the general term and then summing over the range of n. This process often requires the utilization of standard summation formulas for terms like n, n², and constants. The approach typically involves algebraic manipulation and sometimes matching coefficients in order to express the sum of squares in simplified form.

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