Given Data:

Series: \(\displaystyle{6}+{12}+{24}+\ldots+{6144}\)

The common ratio of the series is,

\(\displaystyle{r}={\frac{{{a}_{{2}}}}{{{a}_{{1}}}}}\)

Here, \(\displaystyle{a}_{{1}}\) is the first term of series and \(\displaystyle{a}_{{2}}\) is the second term of the series.

Substitute the values in the above equation.

\(\displaystyle{r}={\frac{{{12}}}{{{6}}}}\)

\(\displaystyle={2}\)

The last term of the series is,

\(\displaystyle{L}={a}_{{1}}{r}^{{{n}-{1}}}\)

Substitute the values in the above equation.

\(\displaystyle{6144}={6}{\left({2}\right)}^{{{n}-{1}}}\)

\(\displaystyle{2}^{{{n}-{1}}}={1024}\)

\(\displaystyle{2}^{{{n}-{1}}}={2}^{{{10}}}\)

\(\displaystyle{n}-{1}={10}\)

\(\displaystyle{n}={11}\)

The sum of finite geometric series is,

\(\displaystyle{S}_{{n}}={\frac{{{a}_{{1}}-{a}_{{1}}{r}^{{n}}}}{{{1}-{r}}}}\)

Here, n is the number of terms in the series.

Susbtitute the values in the above equation.

\(\displaystyle{S}_{{n}}={\frac{{{6}-{6}{\left({2}\right)}^{{{11}}}}}{{{1}-{2}}}}\)

\(\displaystyle{S}_{{n}}={\frac{{{6}-{6}{\left({2048}\right)}}}{{{1}-{2}}}}\)

\(\displaystyle{S}_{{n}}={\frac{{-{12282}}}{{-{1}}}}\)

\(\displaystyle={12282}\)

Thus, the sum of a finite geometric series is 12282.

Series: \(\displaystyle{6}+{12}+{24}+\ldots+{6144}\)

The common ratio of the series is,

\(\displaystyle{r}={\frac{{{a}_{{2}}}}{{{a}_{{1}}}}}\)

Here, \(\displaystyle{a}_{{1}}\) is the first term of series and \(\displaystyle{a}_{{2}}\) is the second term of the series.

Substitute the values in the above equation.

\(\displaystyle{r}={\frac{{{12}}}{{{6}}}}\)

\(\displaystyle={2}\)

The last term of the series is,

\(\displaystyle{L}={a}_{{1}}{r}^{{{n}-{1}}}\)

Substitute the values in the above equation.

\(\displaystyle{6144}={6}{\left({2}\right)}^{{{n}-{1}}}\)

\(\displaystyle{2}^{{{n}-{1}}}={1024}\)

\(\displaystyle{2}^{{{n}-{1}}}={2}^{{{10}}}\)

\(\displaystyle{n}-{1}={10}\)

\(\displaystyle{n}={11}\)

The sum of finite geometric series is,

\(\displaystyle{S}_{{n}}={\frac{{{a}_{{1}}-{a}_{{1}}{r}^{{n}}}}{{{1}-{r}}}}\)

Here, n is the number of terms in the series.

Susbtitute the values in the above equation.

\(\displaystyle{S}_{{n}}={\frac{{{6}-{6}{\left({2}\right)}^{{{11}}}}}{{{1}-{2}}}}\)

\(\displaystyle{S}_{{n}}={\frac{{{6}-{6}{\left({2048}\right)}}}{{{1}-{2}}}}\)

\(\displaystyle{S}_{{n}}={\frac{{-{12282}}}{{-{1}}}}\)

\(\displaystyle={12282}\)

Thus, the sum of a finite geometric series is 12282.