Question

If the $n^{th}$ term of the A.P. $9,7,5,\dots$ is same as the $n^{th}$ term of the A.P. $15,12,9,\dots$ Find $n$.

Answer 1

Step 1: Finding the $n^{th}$ term of the A.P. $9,7,5,\dots$:

Here, $a=9$ and $d=-2$

$$\begin{gathered} a_n=a+\left(n-1\right)d \\ \Rightarrow a_n=9+\left(n-1\right)\left(-2\right) \\ \Rightarrow a_n=9-2n+2 \\ \Rightarrow a_n=11-2n \end{gathered}$$

Step 2: Finding the $n^{th}$ term of the A.P. $15,12,9,\dots$:

Here, $a=15$ and $d=-3$

$$\begin{gathered} a_n=a+\left(n-1\right)d \\ \Rightarrow a_n=15+\left(n-1\right)\left(-3\right) \\ \Rightarrow a_n=15-3n+3 \\ \Rightarrow a_n=18-3n \end{gathered}$$

Step 3: Equating $n^{th}$ term of both the A.P.'s:

$$\begin{gathered} 11-2n=18-3n \\ \Rightarrow 3n-2n=18-11 \\ \Rightarrow n=7 \end{gathered}$$

Hence, the value of $n$ is $7$.