Question

Make the correct alternative in the following question:

If P(n): 49ⁿ + 16ⁿ + $\lambda$ is divisible by 64 for n $\in$ N is true, then the least negative integral value of $\lambda$ is

(a) $-$3 (b) $-$2 (c) $-$1 (d) $-$4

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \mathrm{P}\left(n\right):{49}^n+{16}^n+\lambda \mathrm{is}\mathrm{divisible}\mathrm{by}64\mathrm{for}\mathrm{all}n\in \mathbf{N}. \\ \mathrm{For}n=1, \\ \mathrm{P}\left(1\right)={49}^1+{16}^1+\lambda =65+\lambda \\ \mathrm{As},\mathrm{the}\mathrm{nearest}\mathrm{value}\mathrm{of}\mathrm{P}\left(1\right)\mathrm{which}\mathrm{is}\mathrm{divisible}\mathrm{by}64\mathrm{is}64\mathrm{itself}. \\ \Rightarrow 65+\lambda =64 \\ \Rightarrow \lambda =64-65 \\ \therefore \lambda =-1 \end{gathered}$$

Hence, the correct alternative is option (c).