Question

If the sum of $n$ terms of a GP (with common ratio $r$) beginning with the $p^{th}$ term is $k$ times the sum of an equal number of the same series beginning with the $q^{th}$ term, then the value of $k$ is

• A. $r^{p/q}$
• B. $r^{q/p}$
• C. $r^{p-q}$
• D. $r^{p+q}$

Answer 1

The correct option is C $r^{p-q}$

$p^{th}$ term of the series $=a{r}^{p-1}$ ($a$ is first term)

$q^{th}$ term of th series $=a{r}^{q-1}$

Sum of $n$ term beginning with $p^{th}$ term

$=\frac{a{r}^{p-1}(r^n-1)}{r-1}$

Sum of $n$ term beginning with $q^{th}$ term

$=\frac{a{r}^{q-1}(r^n-1)}{r-1}$

Sum of $n$ term beginning with $p^{th}$ term $=k$ (sum of $n$ term beginning with $q^{th}$ term )

Thus $\frac{a{r}^{p-1}(r^n-1)}{r-1}$$=k\frac{a{r}^{q-1}(r^n-1)}{r-1}$

$\Rightarrow k=\frac{r^{p-1}}{r^{q-1}}$

$\Rightarrow k=r^{p-q}$