Given:
The given expression is [tex]\sum_{1}^{10} 4\left(\frac{1}{4}\right)^{n-1}[/tex]
We need to evaluate the given expression.
Solution:
The given expression is in the form of general form of geometric sequence [tex]a_n=ar^{n-1}[/tex]
The common ratio is [tex]r=\frac{1}{4}[/tex] and the first term is a = 4.
The formula to find the sum of the series is given by
[tex]S_n=a(\frac{1-r^{n}}{1-r})[/tex]
Substituting n= 10, a = 4 and [tex]r=\frac{1}{4}[/tex] , we get;
[tex]S_{10}=4 \cdot \frac{1-\left(\frac{1}{4}\right)^{10}}{1-\frac{1}{4}}[/tex]
[tex]S_{10}=4 \cdot \frac{1-\frac{1}{1048576}}{1-\frac{1}{4}}[/tex]
[tex]S_{10}=4 ( \frac{\frac{1048575}{1048576}}{\frac{3}{4}})[/tex]
[tex]S_{10}=4 ( \frac{1048575}{1048576} \times {\frac{4}{3})[/tex]
[tex]S_{10}= \frac{16777200}{3145728}[/tex]
[tex]S_{10}=5.33[/tex]
Thus, the sum of the 10 terms is 5.33
