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

A. $$\frac{{ab}}{{2\left( {b - a} \right)}}$$

B. $$\frac{{ab}}{{b - a}}$$

C. $$\frac{{3ab}}{{2\left( {b - a} \right)}}$$

D. None of these

Answer: Option C

Solution(By Examveda Team)

First term (a1) = a
Second term (a2) = b
and last term (l) = 2a
∴ d = Second term - First term = b - a
∴ l = an = a + (n - 1)d
$$\eqalign{ & \Rightarrow 2a = a + \left( {n - 1} \right)\left( {b - a} \right) \cr & \Rightarrow \left( {n - 1} \right)\left( {b - a} \right) = a \cr & \Rightarrow n - 1 = \frac{a}{{b - a}} \cr & \Rightarrow n = \frac{a}{{b - a}} + 1 \cr & \Rightarrow n = \frac{{a + b - a}}{{b - a}} \cr & \Rightarrow n = \frac{b}{{b - a}} \cr & \therefore {S_n} = \frac{n}{2}\left[ {a + l} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{b}{{2\left( {b - a} \right)}}\left[ {a + 2a} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{3ab}}{{2\left( {b - a} \right)}} \cr} $$

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