First slide

Arithmetic progression

Question

The sum upto $(2 n + 1)$ terms of the series $a^{2} - (a + d)^{2} + (a + 2 d)^{2} - (a + 3 d)^{2} + \dots$ is

Moderate

A

$a^{2} + 3 n d^{2}$
B

$a^{2} + 2 n a d + n (n - 1) d^{2}$
C

$a^{2} + 3 n a d + n (n - 1) d^{2}$
D

$(a + n d)^{2} + n (n + 1) d^{2}$

Solution

We can write the sum upto $(2 n + 1)$ terms as
$\begin{array}{l} [a + (a + d)] (- d) + [(a + 2 d) + (a + 3 d)] (- d) + \dots \\ [(a + (2 n - 2) d) + (a + (2 n - 1) d] (- d) + (a + 2 n d)^{2} \\ = (- d) [a + (a + d) + (a + 2 d) + \dots \\ + a + (2 n - 1) d] + (a + 2 n d)^{2} \\ = (- d) \frac{2 n}{2} {a + a + (2 n - 1) d} + (a + 2 n d)^{2} \\ \begin{array}{l} = - 2 n a d - n (2 n - 1) d^{2} + a^{2} + 4 n (a d) + 4 n^{2} d^{2} \\ = a^{2} + 2 n a d + n (2 n + 1) d^{2} = (a + n d)^{2} + n (n + 1) d^{2} \end{array} \end{array}$

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