Question:
Make the correct alternative in the following question:
If $P(n): 49^{n}+16^{n}+\lambda$ 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) $-4$
Solution:
We have,
$\mathrm{P}(n): 49^{n}+16^{n}+\lambda$ is divisible by 64 for all $n \in \mathbf{N}$.
For $n=1$,
$\mathrm{P}(1)=49^{1}+16^{1}+\lambda=65+\lambda$
As, the nearest value of $\mathrm{P}(1)$ which is divisible by 64 is 64 itself.
$\Rightarrow 65+\lambda=64$
$\Rightarrow \lambda=64-65$
$\therefore \lambda=-1$
Hence, the correct alternative is option (c).