Question:
Find the sum of all the 11 terms of an AP whose middle most term is 30.
Solution:
Since, the total number of terms (n)=11 [odd]
$\therefore$ Middle most term $=\frac{(n+1)}{2}$ th term $=\left(\frac{11+1}{2}\right)$ th term $=6$ th term
Given that, $a_{6}=30$
$\Rightarrow \quad a+(6-1) d=30$
$\Rightarrow \quad a+5 d=30$ ...(1)
$\because$ Sum of $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$
$\therefore$ $S_{11}=\frac{11}{2}[2 a+(11-1) d]$
$=\frac{11}{2}(2 a+10 d)=11(a+5 d) \quad$ [from Eq. (i)]
$=11 \times 30=330$