If {S_k} denotes the sum of the first k terms | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The sum of the first $k$ terms of an arithmetic progression is given by ${S_k} = \frac{k}{2} \{2a + (k - 1)d\}$.We are given the ratio $\frac{S_{k

Vedclass · Accepted Answer

$2a - d = 0$

Vedclass · Answer

(A) The sum of the first $k$ terms of an arithmetic progression is given by ${S_k} = \frac{k}{2} \{2a + (k - 1)d\}$.We are given the ratio $\frac{S_{kn}}{S_n} = \frac{\frac{kn}{2} \{2a + (kn - 1)d\}}{\frac{n}{2} \{2a + (n - 1)d\}}$.Simplifying the expression,we get:$\frac{S_{kn}}{S_n} = k \left\{ \frac{2a + (kn - 1)d}{2a + (n - 1)d} \right\} = k \left\{ \frac{(2a - d) + knd}{(2a - d) + nd} \right\}$.For this expression to be independent of $n$,the term containing $n$ in the numerator must be proportional to the term containing $n$ in the denominator,or the constant terms must vanish such that $n$ cancels out.If $2a - d = 0$,then the expression becomes:$\frac{S_{kn}}{S_n} = k \left\{ \frac{knd}{nd} \right\} = k^2$.Since $k^2$ is independent of $n$,the condition is $2a - d = 0$.

If ${S_k}$ denotes the sum of the first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively,then ${S_{kn}}/{S_n}$ is independent of $n$ if:

Similar Questions

The difference between the fourth term and the first term of a Geometric Progression is $52.$ If the sum of its first three terms is $26,$ then the sum of the first six terms of the progression is

If the first term of a $G.P.$ is $5$ and the common ratio is $-5$,then which term is $3125$ (in $^{th}$)?

Insert $6$ arithmetic means between $20$ and $76$.

If $\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3 = \frac{k}{21}$,then $k$ equals

If every term of a $G.P.$ with positive terms is the sum of its two previous terms,then the common ratio of the series is

Vedclass Test Series

Exam Paper Generator

Online Exam Module