If {Sₙ} denotes sum of n terms of an arithmetic progression, then value of ({S_{2n}} - {Sₙ}) is

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8. Sequences and Series

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If ${S_n}$ denotes the sum of $n$ terms of an arithmetic progression, then the value of $({S_{2n}} - {S_n})$ is equal to

A

$2{S_n}$

B

${S_{3n}}$

C

$\frac{1}{3}{S_{3n}}$

D

$\frac{1}{2}{S_n}$

Solution

(c) ${S_{2n}} – {S_n} = \frac{{2n}}{2}\{ 2a + (2n – 1)d\} – \frac{n}{2}\{ 2a + (n – 1)d\} $

$ = \frac{n}{2}\{ 4a + 4nd – 2d – 2a – nd + d\} = \frac{n}{2}\{ 2a + (3n – 1)d\} $

$ = \frac{1}{3}.\frac{{3n}}{2}\{ 2a + (3n – 1)d\} = \frac{1}{3}{S_{3n}}$.

Standard 11

Mathematics

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