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-

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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