Answer: The required sum of first terms of the series is \(\dfrac{1705}{64}.\)
Step-by-step explanation: We are given to find the sum of the first five terms of a geometric series with first term and common ratio as follows :
\(a_1=20~~~~~\textup{and}~~~~~r=\dfrac{1}{4}.\)
We know that
the sum of first n terms of a geometric series with first term \(a_1\) and common ratio r is given by
\(S_n=\dfrac{a(1-r^n)}{1-r}.\)
Therefore, the sum of first 5 terms of the given geometric series is given by
\(S_5\\\\\\=\dfrac{a(1-r^5)}{1-r}\\\\\\=\dfrac{20(1-(\frac{1}{4})^5)}{1-\frac{1}{4}}\\\\\\=\dfrac{20\left(1-\frac{1}{1024}\right)}{\frac{3}{4}}\\\\\\=20\times\dfrac{4}{3}\times\dfrac{1023}{1024}\\\\\\=20\times\dfrac{341}{256}\\\\\\=\dfrac{5\times 341}{64}\\\\\\=\dfrac{1705}{64}.\)
Thus, the required sum of first terms of the given geometric series is \(\dfrac{1705}{64}.\)
