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

$-1$

$(49)^n + 16n - 1$
$\Rightarrow (1 + 48)^n + 16n - 1$
$\Rightarrow 1 + 48n + ... 48^n + 16n - 1$
$\Rightarrow 64n + nC_2(48)^2 + nC_3(48)^3 + ... + (48)^n$
$\Rightarrow 64(n + nC_2(6)^2 + nC_3(6)^348 + ... + (6)^n 8^{n-2})$
$\therefore$ $49^n + 16n - 1$ is divisible by $64$

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