Progression (Sequence and Series)HardBloom L3
Question
The first and last two terms of an A.P. are $a,b,c$ respectively in order. Then the sum of all the terms is :
Options
A.$\frac{(2c - a - b)(a + c)}{2(c - b)}$
B.$\frac{(2c - a - b)(a + c)}{(c - b)}$
C.$\frac{(4c - 2a - 2\text{ }b)(a + c)}{(c - b)}$
D.$\frac{(2c + a + b)(c - a)}{(c - b)}$
Solution
Sol. [Key Concept: AP with first/last terms known: number of terms $n = \frac{2c - a - b}{c - b}$, then sum $= \frac{n}{2}(a + c)$.]
$$c = a + (n - 1)(c - b)$$
$$n = \frac{2c - a - b}{c - b}$$
$\text{∑}$ $\text{=}\frac{n}{2}(a + c) = \frac{(2c - a - b)(a + c)}{2(c - b)}$
$$\begin{matrix} & \ \\ & \ \\ \text{~~} & \ = \\ & \ \end{matrix}$$
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