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 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

We have, first term = a, $∴$ T₁ = a
Second term = b, $∴$ T₂ = b
Then common difference d = T₂ – T₁ = b – a
Also, last term = c.
$\begin{array}{l} \Rightarrow c = a + (n - 1) d \Rightarrow n = \frac{c - a + d}{d} . \\ \Rightarrow n = \frac{(b + c - 2 a)}{(b - a)} (∵ d = b - a) \end{array}$
$∴ Sum of n terms S_{n} = \frac{n}{2} (a + l) = \frac{(b + c - 2 a) (a + c)}{2 (b - a)}$

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