Question

If the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, then the 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)

Sum of $n$ terms of A.P with first term = $a$ and common
difference, = $d$ is given by $S_n=\frac{n}{2}[2a+(n-1)d]$
$\therefore S_{10}=5[2a+9d];S_5=\frac{5}{2}[2a+4d]$
According to the given condition,
$\therefore S_{10}=5[2a+9d];S_5=\frac{5}{2}[2a+4d]$
According to the given condition,
$S_{10}=S_5$
$\Rightarrow 5[2a+9d]=4\times \frac{5}{2}[2a+4d]$
$\Rightarrow 2a+9d=2[2a+4d]$
$\Rightarrow 2a+9d=4a+8d$
$\Rightarrow d=2a$
$\Rightarrow \frac{a}{d}=\frac{1}{2}$
$\Rightarrow a:d=1:2$