First slide

Arithmetic progression

Question

If the first, second and the last terms of an A.P. are $a, b, c$ respectively, then the sum of the $A . P .$ is

Moderate

A

$\frac{(a + b) (a + c - 2 b)}{2 (b - a)}$
B

$\frac{(b + c) (a + b - 2 c)}{2 (b - a)}$
C

$\frac{(a + c) (b + c - 2 a)}{2 (b - a)}$
D

none of these

Solution

Let there be $n$ terms in the A.P. Then,
$c = a + (n - 1) (b - a) [∵ d = b - a]$
$\Rightarrow n = \frac{b + c - 2 a}{b - a}$
$∴$ sum of $n$ terms $= \frac{n}{2} (a + c) = \frac{(b + c - 2 a) (a + c)}{2 (b - a)}$

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