Question

If the term of the two AP $9,7,5,\dots$ and $24,21,18,\dots$ are the same, then find the value of $\text{n}$. Also, find that term.

Answer 1

Step 1. Writing the value of $\text{a}$ and $\text{d}$:

Given that $9,7,5,\dots$ and $24,21,18,\dots$ are the AP with the same ${\text{n}}^{th}$term.

We know that ${\text{a}}_n=\text{a}+(\text{n}-1)\text{d}$

$\text{a}=9$ and $\text{d=-2}$ for the first AP

$\text{a}=24$ and $\text{d=-3}$ for the second AP

Step 2. Calculate the value of $\text{n}$ and ${\text{a}}_n$:

$9+(\text{n}-1)(-2)=24+(\text{n}-1)(-3)$

On simplifying the above equation, we get,

$\begin{array}{cc}\Rightarrow & 9-2\text{n}+2=24-3\text{n}+3\\ \Rightarrow & 11-2\text{n}=27-3\text{n}\\ \Rightarrow & \text{n}=16\\ \Rightarrow & {\text{a}}_{\text{n}}=9+15(-2)\\ \Rightarrow & {\text{a}}_n=-21\end{array}$

Therefore, the value of $\text{n}$ is $16$ and the ${16}^{th}$ term is $-21$.