Question

If the sum of the $10$ terms of an A.P. is $4$ times to the sum of its $5$ terms, then the ratio of first term and common difference is:

• A. $1:2$
• B. $2:1$
• C. $2:3$
• D. $3:2$

Answer 1

The correct option is A

$1:2$

Explanation for the correct option.

Sum of $n$ terms of AP, ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{[}\mathbf{2}\mathbf{a}\mathbf{+}\left(n-1\right)\mathbf{d}\mathbf{]}$

Where, $a$ is the first term and $d$ is the common difference.

$\begin{array}{rcl}& \Rightarrow & S_{10}=\frac{10}{2}\left[2a+\left(10-1\right)d\right]\\ & =& 5\left[2a+9d\right]\end{array}$

and

$\begin{array}{rcl}{S}_5& =& \frac{5}{2}\left[2a+\left(5-1\right)d\right]\\ & =& 2.5\left[2a+4d\right]\end{array}$

As per the given information, $S_{10}=4\left(S_5\right)$

$\begin{array}{rcl}& \Rightarrow & 5\left(2a+9d\right)=4\left[2.5\left(2a+4d\right)\right]\\ & \Rightarrow & 10a+45d=20a+40d\\ \Rightarrow 10a& =& 5d\\ \Rightarrow \frac{a}{d}& =& \frac{1}{2}\end{array}$

Therefore, the ratio of first term and common difference is $1:2$.

Hence, option A is correct.