Question

Assertion :If the sum of $n$ terms of a series is $2n^2+3n+1$ then series is in $A.P.$ with common difference 4. Reason: If sum of $n$ terms of an $A.P$ is quadratic expression, then common difference is twice of the coefficient of quadratic term.

• 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 correct but Reason is incorrect
• 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

$\because$ Sum of $n$ terms of an $A.P.$ is

$S_n=\frac{n}{2}[2A+(n-1)D]$

Where $A=$ first term, $D=$ common difference. Hence the sum of $n$ terms of a $A.P.$ is always in the form of quadratic expression & Reason (R) is true

Again $S_n=2n^2+3n+1$

$\therefore t_n=S_n-S_{n-1}$

$=2[n^2-{(n-1)}^2]+3[n-n+1]$

$=2[2n-1]+3=4n+1$

$\therefore D=t_{n+1}-t_n=4$