Question

If the $9^{th}$ term of an A.P be zero, then the ratio of its ${29}^{th}$ and ${19}^{th}$ term is

• A. $1:2$
• B. $2:1$
• C. $1:3$
• D. $3:1$

Answer 1

The correct option is B

$2:1$

Finding the ratio between ${29}^{th}$ and ${19}^{th}$ term:

Step 1 : Find the terms using general term of AP

The formula for the $n\text{th}$ term of an A.P is given as,

${\mathit{a}}_{\mathbf{n}}\mathbf{=}\mathit{a}\mathbf{+}\left(n-1\right)\mathit{d}$, $a_n$ is the $n\text{th}$ term, $a$ is the initial term, $n$ is the number of the term, and $d$ is a common difference.

We know that the $9^{\text{th}}$ term is $0$,

$\begin{array}{rcl}\therefore a_9& =& 0\\ & \Rightarrow & a+\left(9-1\right)d=0\\ & \Rightarrow & a+8d=0\\ & \Rightarrow & a=-8d\end{array}$

Now, the ${29}^{\text{th}}$ term will be,

$\begin{array}{rcl}{a}_{29}& =& a+\left(29-1\right)d\\ & =& a+28d\\ & =& -8d+28d\\ & =& 20d\end{array}$

The ${19}^{\text{th}}$ term will be,

$\begin{array}{rcl}{a}_{19}& =& a+\left(19-1\right)d\\ & =& a+18d\\ & =& -8d+18d\\ & =& 10d\end{array}$

Step 2 : Find the ratio of the terms obtained

The ratio of ${29}^{th}$ and ${19}^{th}$ term is

$\begin{array}{rcl}\frac{a_{29}}{a_{19}}& =& \frac{20d}{10d}\\ & =& \frac{2}{1}\\ & =& 2:1\end{array}$

Hence option (B) is the answer.