Question

Let $s _1, s _2, s _3 \ldots \ldots$. and $t _1, t _2, t _3 \ldots \ldots$ are two arithmetic sequences such that $s _1= t _1 \neq 0 ; s _2=2 t _2$ and $\displaystyle\sum_{i=1}^{10} s_i=\displaystyle\sum_{i=1}^{15} t_i$. 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 $s _1+ s _2+ s _3+\ldots \ldots .+ s _{10}= t _1+ t _2+ t _3+\ldots \ldots+ t _{15}$
let $1^{\text {st }}$ sequence is
let$a_1, a_1+d_1, a_1+2 d_1, \ldots . . . . .$
and $2^{\text {nd }}$ is $a _1, a _1+ d _2, a _1+2 d _2, \ldots \ldots . $ (since $s _1= t _1$ )
given $s_2=2 t_2$
$\therefore a _1+ d _1=2\left( a _1+ d _2\right) $
$\therefore a _1= d _1-2 d _2$ .....(1)
we have to find $\frac{s_2-s_1}{t_2-t_1}=\frac{d_1}{d_2}=$ ?
now $\frac{10}{2}\left[2 a _1+9 d _1\right]=\frac{15}{2}\left[2 a _1+14 d _2\right]$
this gives $ a_1=9 d_1-21 d_2$ .....(2)
from (1) and (2) $ \frac{ d _1}{ d _2}=\frac{19}{8}$