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STD 11 - 8. sequence and series — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 8. sequence and series4 Marks
MCQ
If ${S_k}$ denotes the sum of first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively, then ${S_{kn}}/{S_n}$ be independent of $n$ if
  • ✓
    $2a - d = 0$
  • B
    $a - d = 0$
  • C
    $a - 2d = 0$
  • D
    None of these
✓

Answer

Correct option: A.
$2a - d = 0$
a
(a) $\frac{{{S_{kn}}}}{{{S_n}}} = \frac{{(kn/2)\{ 2a + (kn - 1)d\} }}{{(n/2)\{ 2a + (n - 1)d\} }} = k\left\{ {\frac{{(2a - d) + knd}}{{(2a - d) + nd}}} \right\}$

$i.e.$ if $2a - d = 0$,

then this becomes $\frac{{{k^2}nd}}{{nd}} = {k^2}$ which is obviously independent of $n$.

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