Question

Assertion :The coefficients of $x^n$ in the expansion ${(2x+3)}^n-{(2x+3)}^{n-1}(5-2x)+{(2x+3)}^{n-2}{(5-2x)}^2+...+{(-1)}^n{(5-2x)}^n$ is $(n+1)2^n$ Reason: $a^n+a^{n-1}b+a^{n-2}{b}^2+...+b^n=\frac{a^{n+1}-b^{n+1}}{a-b}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is incorrect but Reason is correct
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

The expansion is a $G.P.$ with $(n+1)$ terms of the from

$a^n+a^{n-1}b+a^{n-2}{b}^2+...+b^n=\frac{a^{n+1}-b^{n+1}}{a-b}=\frac{{(2x+3)}^{n+1}-{(2x-5)}^{n+1}}{8},$

where $a=2x+3$ and $b=2x-5$

$\therefore$ Coefficient of $x^n=\frac{1}{8}[(n+1).2^n\left(3\right)-(n+1).2^n(-5)]=(n+1).2^n$

Therefore, both assertion and reason are and reason is the correct explanation for assertion.