Single Choice

Let $$a_{1},\ a_{2},\ a_{3},\ \ldots,\ a_{100}$$ be an arithmetic progression with $$a_{1}=3$$ and $$S_{p}$$ is sum of 100 terms . For any integer $$n$$ with $$1\leq n \leq 20$$, let $$ m=5n$$. If $$\dfrac{S_{m}}{S_{n}}$$ does not depend on $$n$$, then $$a_{2}$$ is

A$$6$$
B$$7$$
C$$8$$
D$$9$$
Correct Answer

Solution

We know that sum of n terms of an A.P. is given by
$${ S }_{k }=\dfrac { k }{ 2 } \left\{ 2a+(k-1)d \right\} $$

So,$$\displaystyle \dfrac { { S }_{ m } }{ { S }_{ n } } =\frac { { S }_{ 5n } }{ { S }_{ n } }=\dfrac { \dfrac { 5n }{ 2 } \left\{ 6+(5n-1)d \right\} }{ \dfrac { n }{ 2 } \left\{ 6+(n-1)d \right\} } =\frac { 5n\left\{ 6+(5n-1)d \right\} }{ n\left\{ 6+(n-1)d \right\} } =\frac { 5\left\{ (6-d)+5nd \right\} }{ \left\{ (6-d)+n \right\} } $$

Since, $$\displaystyle \frac { { S }_{ m } }{ { S }_{ n } }$$ is independent of n , so $$ d =6$$.

If $$d = 6$$ then $${a}_{2} = {a}_{1} + d = 3 + 6 = 9$$

SIMILAR QUESTIONS

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