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Limits and Derivatives — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits and Derivatives1 Mark
MCQ
if n is a positive ineger then $2^{3n}n - 7n - 1$ is divisible by:
  • A
    $7$
  • B
    $9$
  • ✓
    $49$
  • D
    $81$
✓

Answer

Correct option: C.
$49$
Given, $2^{3 n}-7 n-1=2^{3 \times n}-7 n-1 $
$ =8^n-7 n-1 $
$ =(1+7)^n-7 n-1 $
$ =\left\{{ }^n C_0+{ }^n C_1 {~7}+{ }^n C_2 {~7}^2+\ldots \ldots \ldots+{ }^n C_n {~7}^n\right\}-7 n-1 $
$ =\left\{1+7 n+{ }^n C_2 {~7}^2+\ldots \ldots \ldots+{ }^n C_n {~7}^n\right\}-7 n-1 $
$ ={ }^n C_2 {~7}^2+\ldots \ldots \ldots+{ }^n C_n {~7}^n $
$ =49\left({ }^n C_2+\ldots \ldots \ldots+{ }^n C_n {~7}^{n-2}\right)$
which is divisible by $49$
So, $2^{3n} - 7n - 1$ is divisible by $49$

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