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:
$\frac{L}{p}=\frac{L^{\prime }}{a}=4,\frac{S_n}{S_n^{\prime }}=2$; find out $\frac{d}{q}$ and $\frac{L}{L^{\prime }}$.

Answer 1

$\frac{L}{p}=\frac{L^{\prime }}{a}=4$(given) ..$\left(1\right)$
$\therefore a+(n-1)d=4p$
$p+(n-1)q=4a$
$\therefore \frac{4p-a}{4a-p}=\frac{d}{p}$ ..$\left(2\right)$
$\frac{S_n}{S_n^{\prime }}=\frac{\frac{n}{2}(a+L)}{\frac{n}{2}(p+L^{\prime })}=\frac{a+4p}{p+4a}=2$, given by $\left(1\right)$
$\therefore 2p=7a$ or $\frac{a}{2}=\frac{p}{7}=\lambda$, say $\left(3\right)$
Putting in $\left(2\right),\frac{d}{q}=\frac{28\lambda -2\lambda }{8\lambda -7\lambda }=26$
and $\frac{L}{L^{\prime }}=\frac{p}{a}=\frac{7}{2}$; by $\left(1\right)$ and $\left(3\right)$.