Question

If the sum of the first $11$ terms of an arithmetical progression equals that of the first $19$ terms. Then the sum of its first $30$ terms is

• A. equals to $0$
• B. equals to $-1$
• C. equals to $1$
• D. non unique

Answer 1

The correct option is A equals to $0$

We have to find the sum of first 30 terms, ie, $S_{30}$
Let the first term of the AP is 'a' & common difference is 'd'
So, it is given that
$S_{11}=S_{19}$
$\Rightarrow Sumof11terms=Sumoffirst19terms$
$\Rightarrow \frac{11}{2}[2a+(11-1\left)d\right]=\frac{19}{2}[2a+(19-1\left)d\right]$
$\Rightarrow 22a+11d=38a+342d$
$\Rightarrow -2a=29d$
$\Rightarrow a=-\frac{29d}{2}$
$S_{30}=15(2a+29d)$
put $a=-\frac{29d=29d}{2}$
$S_{30}=15\times [2(-\frac{29d}{2})+29d]$
$=15\times 0$
$=0$