If p(n): 49^n+16^ n is divisible by 64 for n is true, then the le… - Vidyadip

MCQ

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

A
$-3$
B
$-2$
✓
$-1$
D
$-4$

✓

Answer

Correct option: C.

$-1$

$ (49)^n+16 n-1 $
$ \Rightarrow(1+48)^n+16 n-1 $
$ \Rightarrow 1+48 n+\ldots 48^n+16 n-1 $
$ \Rightarrow 64 n+n C_2(48)^2+n C_3(48)^3+\ldots+(48)^n $
$ \Rightarrow 64\left(n+n C_2(6)^2+n C_3(6)^3 48+\ldots+(6)^n 8^{n-2}\right)$
$\therefore 49^n+ 16n - 1$ is divisible by $64$

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