Answer
The sum of the first six terms $\left( {{S}_{6}} \right)$ of the sequence is $1,\frac{1}{10},\frac{1}{100},\frac{1}{1000},\ldots $ is $1.11111$.
Work Step by Step
$1,\frac{1}{10},\frac{1}{100},\frac{1}{1000},\ldots $
As, the sequence is the expansion of ${{a}_{n}}={{\left( 0.1 \right)}^{n}}$, it starts with 1.
So, the expansion of the sequence is,
${{a}_{n}}={{\left( 0.1 \right)}^{n}}$
The values of the first four terms of the sequence are provided:
${{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}}$ = $1,\frac{1}{10},\frac{1}{100},\frac{1}{1000}$
$\begin{align}
& {{a}_{5}}={{\left( 0.1 \right)}^{5}} \\
& =\frac{1}{10000} \\
& =0.0001
\end{align}$
And,
$\begin{align}
& {{a}_{6}}={{\left( 0.1 \right)}^{6}} \\
& =\frac{1}{100000} \\
& =0.00001
\end{align}$
Thus, the value of ${{a}_{5}},{{a}_{6}}$ are $\frac{1}{10000},\frac{1}{100000}$ respectively.
For ${{S}_{6}}$
Sum the first $6$ terms of the sequence,
$\begin{align}
& {{S}_{6}}=1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10000}+\frac{1}{100000} \\
& =1+0.1+0.01+0.001+0.0001+0.00001 \\
& =1.11111
\end{align}$
Thus, the sum of the first six terms, $\left( {{S}_{6}} \right)$, of the sequence is: $1,\frac{1}{10},\frac{1}{100},\frac{1}{1000},\ldots $ is $1.11111$.
