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Sum of n Terms

If the sum of...

Question

If the sum of the first $3 n$ terms is equal to the next $n$ terms of an A.P. whose common difference is non-zero, then reciprocal of ratio of the sum of the first $2 n$ terms to the next $2 n$ terms.

A

\frac{1}{5}

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B

\frac{3}{5}

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C

\frac{2}{4}

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D

\frac{3}{4}

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Solution

The correct option is A $\frac{1}{5}$
Sum of $3 n$ terms $=$ sum of $4 n$ terms $-$ sum of $3 n$ terms.
$\frac{3 n}{2} [2 a + (3 n - 1) d] = \frac{4 n}{2} [2 a + (4 n - 1) d] - \frac{3 n}{2} [2 a + (3 n - 1) d]$
$6 [2 a + (3 n - 1) d] = 4 [2 a + (4 n - 1) d]$
$12 a + 18 n d - 6 d = 8 a + 16 n d - 4 d$
$4 a + 2 n d - 2 d = 0$
$2 a + n d - d = 0$
$2 a = d - n d (1)$
$\frac{s u m 2 n t e r m s}{s u m o f 4 n - s u m o f 2 n t e r m s} = \frac{\frac{2 n}{2} [2 a + (2 n - 1) d]}{\frac{4 n}{2} [2 a + (4 n - 1) d] - \frac{2 n}{2} [2 a + (2 n - 1) d]}$
$⟹ \frac{2 a + 2 n d - d}{4 a + 8 n d - 2 d - 2 a - 2 n d + d} ⟹ \frac{n d}{5 n d} ⟹ \frac{1}{5}$

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