Question

There are two A.P.'s each of $n$ terms $a,a+d,a+2d,...L$ $p,p+q,p+2q,....L$. These A.P.'s satisfy the following conditions: $\frac{L}{P}=\frac{L^{\prime }}{a}=4,\frac{S_n}{S_n}=2;$ find out $\frac{d}{q}and\frac{L}{L^{\prime }}$

• A. $S_1+S_3=2S_2$
• B. $S_1+S_3=S_2$
• C. $S_1+S_3=4S_2$
• D. none of these

Answer 1

The correct option is A $S_1+S_3=2S_2$

Here $a=1.$ for all for and $d=1,2,3,...$ respectively for $S_1,S_{2,}{S}_3...$ and $n=n$ for all
Now, $S_1+S_3=\frac{n}{2}[2.1-(n-1).1]+\frac{n}{2}[2.1+(n-1).3]$
$=\frac{n}{2}[n+1]+\frac{n}{2}[3n-1]=\frac{n}{2}\left[4n\right]={2n}^2.$
${2S}_2=2\left[\frac{n}{2}\right][2.1+(n-1).2]=n\left[2n\right]={2n}^2.$
$\therefore S_1+S_3={2S}_2$ is true