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 $S _{ n }=\frac{ n }{2}[2a +( n -1) d ]$
$\therefore S _{10}=5[2 a +9 d ] ; S _{5}=\frac{5}{2}[2 a +4 d ]$
According to the given condition,
$\therefore S _{10}=5[2 a +9 d ] ; S _{5}=\frac{5}{2}[2 a +4d ]$
According to the given condition,
$S _{10}= S _{5}$
$ \Rightarrow 5[2 a +9 d ]=4 \times \frac{5}{2}[2 a +4 d ]$
$\Rightarrow 2 a +9 d =2[2 a +4 d ]$
$\Rightarrow 2 a +9 d =4 a +8 d $
$\Rightarrow d =2 a$
$\Rightarrow \frac{ a }{ d }=\frac{1}{2}$
$\Rightarrow a : d =1 : 2$