Question

If 9^th term of an A.P. is zero, then its 29^th and 19^th terms are in the ratio __________.

Answer 1

Given:
9^th term (a₉) = 0

Now,
$$\begin{gathered} a_n=a+\left(n-1\right)d \\ {a}_9=a+\left(9-1\right)d \\ \Rightarrow 0=a+8d \\ \Rightarrow a=-8d \end{gathered}$$

Therefore,
$$\begin{gathered} \frac{a_{29}}{a_{19}}=\frac{a+\left(29-1\right)d}{a+\left(19-1\right)d} \\ =\frac{a+28d}{a+18d} \\ =\frac{-8d+28d}{-8d+18d}\left(\because a=-8d\right) \\ =\frac{20d}{10d} \\ =\frac{2}{1} \end{gathered}$$

​Hence, if 9^th term of an A.P. is zero, then its 29^th and 19^th terms are in the ratio 2 : 1.