Question

If an AP, ratio of the $4$th and $9$th terms is $1:3$, find the ratio of $12$ term and $5$th term?

Answer 1

$\frac{a_4}{a_9}=\frac{1}{3}\frac{a+3d}{a+8d}=\frac{1}{3}3a+9d=a+8d2a=-da=\frac{-d}{2}$

Now $\frac{a_{12}}{a_5}=\frac{a+11d}{a+4d}=\frac{\frac{-d}{2}+11d}{\frac{-d}{2}+4d}$

$=\frac{\frac{-d+22d}{2}}{\frac{-d+8d}{2}}$

$=\frac{\frac{21d}{2}}{\frac{7d}{2}}$

$=\frac{21}{7}$

$=3:1$.