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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. $\frac{a b}{2(b-a)}$ B. $\frac{a b}{b-a}$ C. $\frac{3 a b}{2(b-a)}$ D. None of these Answer: Option C
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Solution(By Apex Team)

First term ($\mathrm{a}_{1}$) = a Second term ($\mathrm{a}_{2}$) = b and last term (l) = 2a ∴ d = Second term – First term = b – a $\begin{aligned}&\therefore l=a_n=a+(n-1)d\\ &\Rightarrow2a=a+(n-1)(b-a)\\ &\Rightarrow(n-1)(b-a)=a\\ &\Rightarrow n-1=\frac{a}{b-a}\\ &\Rightarrow n=\frac{a}{b-a}+1\\ &\Rightarrow n=\frac{a+b-a}{b-a}\\ &\Rightarrow n=\frac{b}{b-a}\\ \therefore S_n&=\frac{n}{2}[a+l]\\ &\quad=\frac{b}{2(b-a)}[a+2a]\\ &\quad=\frac{3ab}{2(b-a)}\end{aligned}$

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