Question

The sum of the series \[5.05+1.212+0.29088+...\infty \] is
A. \[6.93378\]
B. \[6.87342\]
C. \[6.74384\]
D. \[6.64474\]

Answer 1

Hint: In this question, we are to find the sum of the terms of the given series. If we find the type of the series i.e., the given series is arithmetic or geometric, we are able to find the sum of the series using the appropriate formula.



Formula Used:The sum of the infinite terms in the G.P series is calculated by
 ${{S}_{\infty }}=\frac{a}{1-r}$ where $r=\frac{{{a}_{n}}}{{{a}_{n-1}}}$
Here ${{S}_{\infty }}$ is the sum of the infinite terms of the series; $a$ is the first term in the series, and $r$ is the common ratio.



Complete step by step solution:The given series is \[5.05+1.212+0.29088+...\infty \]
To find the type of progression, calculate the change between two consecutive terms as below:
The common ratio is calculated by
$r=\frac{{{a}_{n}}}{{{a}_{n-1}}}$
So,
\[\begin{align}
  & {{r}_{1}}=\frac{1.212}{5.05} \\
 & \text{ }=0.24 \\
 & {{r}_{2}}=\frac{0.29088}{1.212} \\
 & \text{ }=0.24 \\
\end{align}\]
Since both ratios are the same, the series is a geometric series with a common ratio \[r=0.24\].
Thus, the sum of the infinite terms of the series is calculated by the formula,
${{S}_{\infty }}=\frac{a}{1-r}$
On substituting $a=5.05;r=0.24$, we get
\[\begin{align}
  & {{S}_{\infty }}=\frac{5.05}{1-0.24} \\
 & \text{ }=\frac{5.05}{0.76} \\
 & \text{ }=6.64474 \\
\end{align}\]



Option ‘D’ is correct

Note: Here the given series is in G.P. So, using the sum of infinite terms formula, the required value is calculated.