In the arithmetic progression whose common difference is non-zero… - Vidyadip

MCQ

In the arithmetic progression whose common difference is non$-$zero, the sum of first $3n$ terms is equal to the sum of next $n$ terms. Then the ratio of the sum of the first $2n$ terms to the next $2n$ terms is:

✓
$\frac{1}{5}$
B
$\frac{2}{3}$
C
$\frac{3}{4}$
D
None of these

✓

Answer

Correct option: A.

$\frac{1}{5}$

$\text{S}_{3\text{n}}=\text{S}_{4\text{n}}-\text{S}_{3\text{n}}$
$\Rightarrow2\text{S}_{3\text{n}}=\text{S}_{4\text{n}}$
$\Rightarrow2\times\frac{3\text{n}}{2}\{2\text{a}+(3\text{n}-1)\text{d}\}=\frac{4\text{n}}{2}\{2\text{a}+(4\text{n}-1)\text{d}\}$
$\Rightarrow3\{2\text{a}+(3\text{n}-1)\text{d}\}=2\{2\text{a}+(4\text{n}-1)\text{d}\}$
$\Rightarrow6\text{a}+9\text{nd}-3\text{d}=4\text{a}+8\text{nd}-2\text{d}$
$\Rightarrow2\text{a}+\text{nd}-\text{d}=0$
$\Rightarrow2\text{a}+(\text{n}-1)\text{d}=0\ .....(1)$
Required ratio: $\frac{\text{S}_{2\text{n}}}{\text{S}_{4\text{n}}-\text{S}_{2\text{n}}}$
$\frac{\text{S}_{2\text{n}}}{\text{S}_{4\text{n}}-\text{S}_{2\text{n}}}=\frac{\frac{2n}{2}\{2a+(2\text{n}-1)\text{d}\}}{\frac{\text{n}4}{2}\{2\text{a}+(4\text{n}-1)\text{d}\}-\frac{2\text{n}}{2}\{2\text{a}+(2\text{n}-1)\}\text{d}}$
$=\frac{\text{n}(\text{nd})}{2\text{n}(3\text{nd})-\text{n}(\text{nd})}$
$=\frac{1}{6-1}$
$=\frac{1}{5}$

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