Question

Assertion :If sum of $n$ terms of two arithmetic progressions are in the ratio $(3n+8):(7n+15)$ then ratio of their $n^{th}$ term is $3:16$ Reason: If $S_n$ is quadratic expression, then $t_n=S_n-S_{n-1}$

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

Given $\frac{S_n}{S_n{}^{\prime }}=\frac{3n+8}{7n+15}=\frac{n(3n+8)}{n(7n+15)}=\frac{3n^2+8n}{7n^2+15n}$

$\Rightarrow \frac{t_n}{t_n{}^{\prime }}=\frac{S_n-S_{n-1}}{S_n{}^{\prime }-S_{n-1}{}^{\prime }}=\frac{(3n^2+8n)-3{(n-1)}^2-8(n-1)}{7n^2+15n-7{(n-1)}^2-15(n-1)}$

$\Rightarrow \frac{t_n}{t_n{}^{\prime }}=\frac{6n+5}{14n+8}$

$\Rightarrow t_n:t_n{}^{\prime }=6n+5:14n+8$

Now Assertion (A) is false & Reason (R) is correct.