Each of these questions contains two statements, Statement 1(Assertion) and Statement 2(Reason).
State which of the following is correct :
(A) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I.
(B) Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I
(C) Statement-I is true, Statement-II is false
(D) Statement-I is false, Statement-II is true
Statement -I : The sum of $n$ terms of two arithmetic progressions are in the ratio $(7 n + 1) : (4 n + 17)$ , then the ratio of their $n^{t h}$ terms is $7 : 4$ .
Statement-II : If $S_{n} = a n^{2} + b n + c,$ then $T_{n} = S_{n} - S_{n - 1}$

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Solution

The correct option is D D
Statement 1:
$\frac{S_{n}}{S_{n}^{^{'}}} = \frac{7 n + 1}{4 n + 17} = \frac{\frac{n}{2} [7 (n - 1) + 7 + 1]}{\frac{n}{2} [4 (n - 1) + 4 + 17]}$
$= \frac{\frac{n}{2} [2 * 4 + 7 (n - 1)]}{\frac{n}{2} [2 * 10.5 + 4 (n - 1)]}$

$\frac{T_{n}}{T_{n}^{^{'}}} = \frac{4 + 7 (n - 1)}{10.5 + 4 (n - 1)} \neq \frac{7}{4}$

Statement 1 $\to$ False

Statement 2:
Whatever is $S_{n}$ , $T_{n} = S_{n} - S_{n - 1}$ is always true
Statement 2 $\to$ True

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