Question

Assertion :If the sequence $\left\{t_n\right\}$ is monotonically increasing ihen the sequence $\{\sum t_n\},$ Where $\sum t_n=\frac{t_1+t_2+...+t_n}{n}$ is also monotonically increasing Reason: If $\left\{t_n\right\}$ is monotonically increasing then the expression $\sum t_{n+1}-\sum t_n$ is positive

• 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

$n\sum t_n=t_1+t_2+....+t_n$
$\therefore (n+1)\sum t_{n+1}=t_1+t_2+....+t_n+t_{n+1}$
$\sum t_{n+1}-\sum t_n=\frac{t_1+t_2+....+t_{n+1}}{n+1}-\frac{t_1+t_2+....+t_n}{n}$
$=\frac{n(t_1+t_2+....+t_{n+1})-(n+1)(t_1+t_2+....+t_n)}{n(n+1)}$
$=\frac{n{t}_{n+1}-(t_1+t_2+....+t_n)}{n(n+1)}$
$=\frac{(t_{n+1}-t_1)+(t_{n+1}-t_2)+....+(t_{n+1}-t_n)}{n(n+1)}>0$
Since, $t_{n+1}>t_1,t_2....t_n$
Therefore, $\sum t_n$ is monotonically increasing.