First slide

Arithmetic progression

Question

If the first, second and last terms of an A.P. are a, b and 2a respectively, then its sum is

Moderate

A

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

$\frac{ab}{b - a}$
C

$\frac{3 ab}{2 (b - a)}$
D

None of these

Solution

Here a₁ = a and a₂ = b
$∴$ Common difference d = a₂ – a₁ = b – a
Let n be the number of terms in the series
$\begin{array}{l} ∴ a_{n} = 2 a = a + (n - 1) d \\ or (n - 1) d = a or (n - 1) (b - a) = a \\ ∴ n - 1 = \frac{a}{b - a} \\ or n = \frac{a}{b - a} + 1 = \frac{a + b - a}{b - a} = \frac{b}{b - a} \\ ∴ Sum = \frac{n}{2} (a_{1} + a_{n}) = \frac{b}{2 (b - a)} (a + 2 a) = \frac{3 ab}{2 (b - a)} . \end{array}$

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