Question

The sum of first ten terms of an A.P. is four times the sum of its first five terms. What is the ratio of first term and common difference?

• A. 1
• B. $\frac{1}{2}$
• C. 4
• D. $\frac{1}{4}$

Answer 1

The correct option is B

$\frac{1}{2}$

Let $S_{10}$ be the sum of first 10 terms and $S_5$ be the sum of first 5 terms.
The sum upto $n$ terms of an AP of first term $a$ and common difference $d$ is given by
$S_n=\frac{n}{2}[2a+(n-1\left)d\right].$

Given, $S_{10}=4S_5.$

$\Rightarrow \frac{10}{2}[2a+(10-1\left)d\right]=4\times \frac{5}{2}[2a+(5-1\left)d\right]$

$\Rightarrow \frac{10}{2}[2a+9d]=4\times \frac{5}{2}[2a+4d]$
$\Rightarrow 2a+9d=4a+8d$
$\Rightarrow 2a=d$
$\Rightarrow \frac{a}{d}=\frac{1}{2}$