Question

If n – 2, 4n – 1, and 5n + 2 are in A.P., then n = _______.

Answer 1

If n – 2, 4n – 1, and 5n + 2 are in A.P.,
then their common difference must be same.

$$\begin{gathered} \therefore a_2-a_1=a_3-a_2=d \\ \Rightarrow \left(4n-1\right)-\left(n-2\right)=\left(5n+2\right)-\left(4n-1\right) \\ \Rightarrow 4n-1-n+2=5n+2-4n+1 \\ \Rightarrow 3n+1=n+3 \\ \Rightarrow 3n-n=3-1 \\ \Rightarrow 2n=2 \\ \Rightarrow n=1 \end{gathered}$$

Hence, n = 1.