First slide

Arithmetic progression

Question

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

Moderate

A

$c^{2} n {(n + 1)}^{2}$
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

Solution

$\begin{array}{l} \begin{aligned} If t_{r} = S_{r} - S_{r - 1} \\ = c r (r - 1) - c (r - 1) (r - 2) \\ = c (r - 1) (r - r + 2) = 2 c (r - 1) \end{aligned} \\ We have, \\ t_{1}^{2} + t_{2}^{2} + \dots \dots \dots + t_{n}^{2} = 4 c^{2} (0^{2} + 1^{2} + 2^{2} + \dots \dots \dots + (n - 1)^{2}) \\ \begin{aligned} = 4 c^{2} \frac{(n - 1) n (2 n - 1)}{6} \\ = \frac{2}{3} c^{2} n (n - 1) (2 n - 1) \end{aligned} \end{array}$

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