Question

Make the correct alternative in the following question:

If ${10}^n+3\times 4^{n+2}+\lambda$ is divisible by 9 for all n $\in$ N, then the least positive integral value of $\lambda$ is

(a) 5 (b) 3 (c) 7 (d) 1

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{P}\left(n\right):{10}^n+3\times 4^{n+2}+\lambda \mathrm{be}\mathrm{divisible}\mathrm{by}9\mathrm{for}\mathrm{all}n\in \mathbf{N}. \\ \mathrm{For}n=1, \\ \mathrm{P}\left(1\right)={10}^1+3\times 4^{1+2}+\lambda \\ =10+3\times 4^3+\lambda \\ =10+192+\lambda \\ =202+\lambda \\ \mathrm{As},\mathrm{the}\mathrm{least}\mathrm{value}\mathrm{of}\mathrm{P}\left(1\right)\mathrm{which}\mathrm{is}\mathrm{divisible}\mathrm{by}9\mathrm{is}207. \\ \Rightarrow 202+\lambda =207 \\ \Rightarrow \lambda =207-202 \\ \therefore \lambda =5 \end{gathered}$$

Hence, the correct alternative is option (a).