Question

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

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true

Answer 1

Answer: (D)

$\because \frac{S_n}{S_n^{\prime }}=\frac{3n+8}{7n+15}=\frac{n\left(3n+8\right)}{n\left(7n+15\right)}$
$=\frac{3n^2+8n}{7n^2+15n}$
$\Rightarrow \frac{t_n}{t_n^{\prime }}=\frac{S_n-S_{n-1}}{S_n^{\prime }-S_{n-1}^{\prime }}$
$=\frac{\left(3n^2+8n\right)-3{\left(n-1\right)}^2-8\left(n-1\right)}{7n^2+15n-7{\left(n-1\right)}^2-15\left(n-1\right)}$
$\Rightarrow \frac{t_n}{t_n^{\prime }}=\frac{6n+5}{14n+8}$
$\Rightarrow t_n:t_n^{\prime }=6n+5:14n+8$
Now Assertion (A) is false & Reason (R) is correct.