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)

$\because \frac{S_{n}}{S'_{n}} = \frac{3n +8}{7n +15} = \frac{n \left(3n +8\right)}{n \left(7n +15\right)}$
$= \frac{3n^{2} +8n}{7n^{2} +15n}$
$\Rightarrow \frac{t_{n}}{t'_{n}} = \frac{S_{n} -S_{n-1}}{S'_{n} -S'_{n-1}}$
$= \frac{\left(3n^{2} +8n\right)-3\left(n -1\right)^{2} -8\left(n -1\right)}{7n^{2} +15n -7\left(n -1\right)^{2} -15\left(n -1\right)}$
$\Rightarrow \frac{t_{n}}{t'_{n}} = \frac{6n +5}{14n +8}$
$\Rightarrow t_{n} :t'_{n} = 6n +5 :14n +8$
Now Assertion (A) is false & Reason (R) is correct.