Question

Let $s_1,s_2,s_3\dots \dots$. and $t_1,t_2,t_3\dots \dots$ are two arithmetic sequences such that $s_1=t_1\ne 0;s_2=2t_2$ and $\sum _{i=1}^{10}{s}_i=\sum _{i=1}^{15}{t}_i$. Then the value of $\frac{s_2-s_1}{t_2-t_1}$ is

• A. $8/3$
• B. $3/2$
• C. $19/8$
• D. 2

Answer 1

Answer: (C)

Given $s_1+s_2+s_3+\dots \dots .+s_{10}=t_1+t_2+t_3+\dots \dots +t_{15}$
let $1^{\text{st }}$ sequence is
let$a_1,a_1+d_1,a_1+2d_1,\dots .....$
and $2^{\text{nd }}$ is $a_1,a_1+d_2,a_1+2d_2,\dots \dots .$ (since $s_1=t_1$ )
given $s_2=2t_2$
$\therefore a_1+d_1=2\left(a_1+d_2\right)$
$\therefore a_1=d_1-2d_2$ .....(1)
we have to find $\frac{s_2-s_1}{t_2-t_1}=\frac{d_1}{d_2}=$ ?
now $\frac{10}{2}\left[2a_1+9d_1\right]=\frac{15}{2}\left[2a_1+14d_2\right]$
this gives $a_1=9d_1-21d_2$ .....(2)
from (1) and (2) $\frac{d_1}{d_2}=\frac{19}{8}$