Question

If the n^th terms of two A.P.s: 9,7,5.... and 24, 21, 18, ... are the same, then the value of n is _________.

Answer 1

The given A.P.s are:
9,7,5.... and 24, 21, 18, ...

9, 7, 5,...
First term (a) = 9
Common difference (d) = 7 − 9 = −2
$$\begin{gathered} n^{\mathrm{th}}\mathrm{term}\left(a_n\right)=a+\left(n-1\right)d \\ =9+\left(n-1\right)\left(-2\right) \\ =9-2n+2 \\ =11-2n...\left(1\right) \end{gathered}$$

24, 21, 18, ...
First term (a) = 24
Common difference (d) = 21 − 24 = −3
$$\begin{gathered} n^{\mathrm{th}}\mathrm{term}\left(b_n\right)=a+\left(n-1\right)d \\ =24+\left(n-1\right)\left(-3\right) \\ =24-3n+3 \\ =27-3n...\left(2\right) \end{gathered}$$

It is given that, a_n = b_n
$$\begin{gathered} \Rightarrow 11-2n=27-3n \\ \Rightarrow 3n-2n=27-11 \\ \Rightarrow n=16 \end{gathered}$$

Hence, the value of n is 16.