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Question: If \(S _ { n }\) denotes the sum of \(n\) terms of an arithmetic progression, then the value of \...

If SnS _ { n } denotes the sum of nn terms of an arithmetic progression, then the value of (S2n−Sn)\left( S _ { 2 n } - S _ { n } \right) is equal to.

A

2Sn2 S _ { n }

B

S3nS _ { 3 n }

C

13S3n\frac { 1 } { 3 } S _ { 3 n }

D

12Sn\frac { 1 } { 2 } S _ { n }

Answer

13S3n\frac { 1 } { 3 } S _ { 3 n }

Explanation

Solution

S2n−Sn=2n2{2a+(2n−1)d}−n2{2a+(n−1)d}S _ { 2 n } - S _ { n } = \frac { 2 n } { 2 } \{ 2 a + ( 2 n - 1 ) d \} - \frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}

=n2{4a+4nd−2d−2a−nd+d}=n2{2a+(3n−1)d}= \frac { n } { 2 } \{ 4 a + 4 n d - 2 d - 2 a - n d + d \} = \frac { n } { 2 } \{ 2 a + ( 3 n - 1 ) d \}

=13⋅3n2{2a+(3n−1)d}=13S3n= \frac { 1 } { 3 } \cdot \frac { 3 n } { 2 } \{ 2 a + ( 3 n - 1 ) d \} = \frac { 1 } { 3 } S _ { 3 n } .