Respuesta :
The sum of the series is [tex]S_{6}=25.5936[/tex]
Explanation:
The series is [tex]32-8+2-0.5+\ldots[/tex]
First let us find the common difference r,
[tex]r=\frac{-8}{32}\\r=\frac{-1}{4}[/tex]
Thus, the common difference is [tex]r=-\frac{1}{4}[/tex]
The formula to find the sum of the first 6 terms is given by
[tex]S_{n}=a_{1} \cdot \frac{1-r^{n}}{1-r}[/tex]
where [tex]a_{1}=32[/tex] , [tex]r=-\frac{1}{4}[/tex] and [tex]n=6[/tex], we get,
[tex]S_{6}=32 \cdot \frac{1-\left(-\frac{1}{4}\right)^{6}}{1-\left(-\frac{1}{4}\right)}[/tex]
Simplifying, we get,
[tex]\begin{aligned}S_{6} &=32 \cdot \frac{1-0.000244}{1.25} \\&=32 \cdot \frac{0.999756}{1.25}\end{aligned}[/tex]
Dividing, we get,
[tex]S_{6}=32(0.7998)[/tex]
Multiplying we have,
[tex]S_{6}=25.5936[/tex]
Thus, the sum of the first 6 terms of the series is [tex]S_{6}=25.5936[/tex]
