What is the sum of first 3n term of an AP , if the sunm of first n term is 2n and sum of first 2n term is 5n


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Question Number 61137 by Tawa1 last updated on 29/May/19

$What is the sum of first 3 n term of an AP, if the sunm of first n term is$ $2 n and sum of first 2 n term is 5 n$
	Answered by Kunal12588 last updated on 29/May/19

	$S_{n} = 2 n$ $\Rightarrow \frac{n}{2} (2 a + (n - 1) d) = 2 n$ $\Rightarrow 2 a + n d - d - 4 = 0$ $S_{2 n} = 5 n$ $\Rightarrow \frac{2 n}{2} (2 a + (2 n - 1) d) = 5 n$ $\Rightarrow 2 a + 2 n d - d - 5 = 0$ $s u b t r a c t i n g t h e r e d o n e s$ $n d - 1 = 0$ $\Rightarrow n d = 1$ $p u t t i n g v a l u e o f n d i n e i t h e r o f r e d l i n e s$ $2 a - d - 3 = 0$ $\Rightarrow 2 a - d = 3$ $S_{3 n} = \frac{3 n}{2} (2 a + (3 n - 1) d)$ $= \frac{3 n}{2} (2 a - d + 3 n d)$ $u s i n g b l u e o n e s$ $S_{3 n} = \frac{3 n}{2} (3 + 3) = 9 n$
	Commented by Kunal12588 last updated on 29/May/19

	$S_{p n} = \frac{p n}{2} (2 a + (p n - 1) d) = \frac{p (p + 3) n}{2}; p \in N$
	Commented by Tawa1 last updated on 29/May/19

	$God bless you sir$
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