Question

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

Answer 1

$\frac{L}{p}=\frac{L^{\prime }}{a}=4$ (given)...(1)

$a+(n-1)d=4p$

$p+(n-1)q=4a$

$\frac{4p-a}{4a-p}=\frac{d}{q}$...(2)

$\frac{S_n}{{S_n}^{\prime }}=\frac{\frac{n}{2}(a+L)}{\frac{n}{2}(p+L^{\prime })}=\frac{a+4p}{p+4a}=2$ given by (1)

$2p=7aor\frac{a}{2}=\frac{p}{7}=\lambda$ ...(3)

Putting in (2)

$ \dfrac { d }{ q } =\dfrac {

28\lambda -2\lambda }{ 8\lambda -7\lambda } =26$ and

$\frac{L}{L^{\prime }}=\frac{p}{a}=\frac{7}{2}$ by (1) and (3)

$2\frac{L}{L^{\prime }}=7$