Question

If $S _ { k }$ denotes the sum of first $k$terms of an arithmetic progression whose first term and common difference are $a$ and respectively, then $S _ { k n } / S _ { n }$ be independent of $n$ if.

• A. $2 a - d = 0$
• B. $a - d = 0$
• C. $a - 2 d = 0$
• D. None of these

Answer 1

Answer: (A)

$2 a - d = 0$

Answer 2

$\frac { S _ { k n } } { S _ { n } } = \frac { ( k n / 2 ) \{ 2 a + ( k n - 1 ) d \} } { ( n / 2 ) \{ 2 a + ( n - 1 ) d \} } = k \left\{ \frac { ( 2 a - d ) + k n d } { ( 2 a - d ) + n d } \right\}$

i.e. if $2 a - d = 0$, then this becomes $\frac { k ^ { 2 } n d } { n d } = k ^ { 2 }$ which is obviously independent of $n$.