Question

The ratio of $7$th to $3$rd term of an A.P. is $12:5$. Find the ratio of $13$th to $4$th term.

Answer 1

We know that the formula for the nth term is $t_n=a+(n-1)d$, where $a$ is the first term, $d$ is the common difference.

The $3$rd, $4$th,$7$th and $13$th term of an A.P are:

$t_3=a+(3-1)d=a+2d$

$t_4=a+(4-1)d=a+3d$

$t_7=a+(7-1)d=a+6d$

$t_{13}=a+(13-1)d=a+12d$

It is given that the ratio of $7$th to $3$rd term is $12:5$, therefore,

$\frac{a+6d}{a+2d}=\frac{12}{5}\Rightarrow 5(a+6d)=12(a+2d)\Rightarrow 5a+30d=12a+24d\Rightarrow 12a-5a=30d-24d\Rightarrow 7a=6d\Rightarrow a=\frac{6}{7}d......\left(1\right)$

Therefore, using $a=\frac{6}{7}d$, the ratio of $13$th to $4$th term is

$\frac{a+12d}{a+3d}=\frac{\frac{6}{7}d+12d}{\frac{6}{7}d+3d}=\frac{\frac{6d+84d}{7}}{\frac{6d+21d}{7}}=\frac{90d}{27d}=\frac{10}{3}=10:3$

Hence, the ratio of $13$th to $4$th term is$10:3$.