Answer
3066
Work Step by Step
The provided series $6+12+24+\cdots $.
Here, ${{a}_{1}}=6$, $n=9$ and
$\begin{align}
& r=\frac{12}{6} \\
& =2
\end{align}$
Substituting ${{a}_{1}}=6$, $n=9$ and $r=2$ in ${{S}_{n}}=\frac{{{a}_{1}}\left( 1-{{r}^{n}} \right)}{1-r}$ , for any $r\ne 1$, we get
$\begin{align}
& {{S}_{9}}=\frac{6\left( 1-{{2}^{9}} \right)}{1-2} \\
& =\frac{6\left( 1-512 \right)}{-1} \\
& =\frac{6\left( -511 \right)}{-1} \\
& =3066
\end{align}$
Thus, the sum of the first nine terms, ${{S}_{9}}$ for $6+12+24+\cdots $ is 3066.
