First slide

Arithmetic progression

Question

In an A.P of which $a$ is the first , if the sum of the first $p$ terms is zero, then the sum of the next $q$ terms is

Moderate

A

$\frac{- a (p + q) p}{q + 1}$
B

$\frac{a (p + q) p}{p + 1}$
C

$\frac{- a (p + q) q}{p - 1}$
D

none of these

Solution

$\begin{array}{l} Given S_{p} = 0, therefore, \\ \frac{p}{2} [2 a + (p - 1) d] = 0 \Rightarrow d = \frac{- 2 a}{p - 1} \end{array}$
$\begin{array}{l} Sum of next q terms is sum of an A.P whose first term will be \\ T_{p + 1} = a + p d \end{array}$
$\begin{array}{l} ∴ S = \frac{q}{2} [2 (a + p d) + (q - 1) d] \\ = \frac{q}{2} [2 a + (p - 1) d + (p + q) d] \\ = \frac{q}{2} [0 - (p + q) \frac{2 a}{p - 1}] \\ = - a \frac{(p + q) q}{p - 1} \end{array}$

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