Question

If the sum of the first ten terms of an $A.P$ is four times the sum of its first five terms, then ratio of the first term to the common difference is

• A. $1:2$
• B. $2:1$
• C. $1:4$
• D. $4:1$

Answer 1

Answer: (A)

$1:2$

From the given statement we can conclude

$S_{10}=4S_5$

$\Rightarrow \frac{10}{2}[2a_1+9d]=4\left(\frac{5}{2}\right)[2a_1+4d]$

$\Rightarrow 10(2a_1+9d)=20(2a_1+4d)$

$\Rightarrow 2a_1+9d=2(2a_1+4d)$

$\Rightarrow 2a_1+9d=4a_1+8d$

$\Rightarrow 9d-8d=2a_1$

$\Rightarrow d=2a_1$

$\frac{a_1}{d}=\frac{1}{2}$.

Hence the ratio is $1:2$.