Question

Assertion (A) : $2^{4n}-2^n(7n+1)$ is divisible by the square of $14$, where $n$ is a natural number.
Reason (R) : $(1+x{)}^n=1{+}^n{C}_{1}x+...{+}^n{C}_n{x}^n\forall n\epsilon N.$

• A. Both A and R are individually true and R is the correct explanation of A.
• B. Both A and R are individually true and R is not correct explanation of A.
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (A)

Both A and R are individually true and R is the correct explanation of A.

$2^{4n}-2^n(7n+1)$
$=2^n[2^{3n}-7n-1]$
$=2^n[8^n-7n-1]$
$=2^n\left[\right(1+7{)}^n-7n-1]$
$=2^n\left[\right(1+7n+{}^n{C}_2{7}^2+{}^n{C}_3{7}^3...{}^n{C}_n{7}^n)-1-7n]$
$=2^n({}^n{C}_2{7}^2+{}^n{C}_3{7}^3...{}^n{C}_n{7}^n)$
$=7^2{2}^n({}^n{C}_2+{}^n{C}_3{7}^{.}..{}^n{C}_n{7}^{n-2})$
$=\left(14{)}^2{2}^{n-2}\right({}^n{C}_2+{}^n{C}_3{7}^{.}..{}^n{C}_n{7}^{n-2})$ ...($n\ge 2$)
Hence both assertion and reason are true and R is the correct explanation of A