Mathematics - Sequence & Series Question with Solution | TestHub

Let $a_1$ be the first term and $d$ be the common difference of $A.P.$

 $\therefore {\mathrm{S}}_{11}=0 \Rightarrow \frac{11}{2}\left(2a_1+10d\right)=0$

$\Rightarrow {\mathrm{a}}_1+5\mathrm{d}=0$

 Now $a_1+a_3+a_5+\dots .+a_{23}=\frac{12}{2}\left(2a_1+\left(12-1\right)2d\right)$ $=12\left(a_1+11d\right)$
$=12\left(a_1-\frac{11a_1}{5}\right)$
$=-\frac{72a_1}{5}$