The first and last terms of an A.P. are a and l respectively. If s is the sum of all the terms of the A.P., then the common difference of the A.P. is

The first and last terms of an A.P. are a and l respectively. If s is the sum of all the terms of the A.P., then the common difference of the A.P. is

A
$\frac{l^{2} - a^{2}}{s - (l + a)}$
B
$\frac{l^{2} - a^{2}}{2 s - (l + a)}$
C
$\frac{l^{2} + a^{2}}{2 s - (l + a)}$
D
$\frac{l^{2} + a^{2}}{s - (l - a)}$

Solution:

Let n be the number of terms. Then

$s = \frac{n}{2} (a + l) \Rightarrow n = \frac{2 s}{a + l}$

Also,

$d = \frac{l - a}{n - 1} = \frac{(l - a) (l + a)}{2 s - (l + a)} = \frac{l^{2} - a^{2}}{2 s - (l + a)}$

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