If {Sₖ} denotes sum of first k terms of an arithmetic progression whose first term and common diff

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8. Sequences and Series

hard

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

$2a - d = 0$

$a - d = 0$

$a - 2d = 0$

None of these

Solution

(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 $n$.

Standard 11

Mathematics

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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

Solution

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