The value for S4 is represented by the third option, [tex]\frac{208}{375}[/tex] .
Geometric Sequences
You can define a geometric sequence when you have a common ratio between the numbers of a sequence. This common ratio can be found when you multiply or divide the previous term by the next term of the sequence and you find the same value. The formula for geometric sequences is:
[tex]a_n=a_1*r^{n-1}[/tex], where:
[tex]a_n[/tex]=[tex]n^{th}[/tex] term
[tex]a_1[/tex]= the first term
Example
2, 4,8, 16, ....
Here, you have:
[tex]a_1[/tex]= 2
[tex]r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=2[/tex] , then: 2 x 2= 4; 4 x 2=8; 8 x 2=16 - Note that the numbers represent the example sequence.
For solving this exercise also it is necessary to apply the formula for the geometric sequence for finite terms since the question asks [tex]S_4[/tex], in the other words, the sum for n=4. Thus,[tex]S_n=\frac{a_1*(1-r^n)}{1-r}[/tex].
For the given series, when n=4, you have
[tex]\frac{2}{3} *(\frac{-1}{5})^{n-4} \\ \\ \frac{2}{3} *(\frac{-1}{5})^{4-4}\\ \\ \frac{2}{3} *(\frac{-1}{5})^{0}\\ \\ \frac{2}{3} *1=\frac{2}{3}[/tex]
Therefore, [tex]a_1=\frac{2}{3}[/tex].
Now, you will find S4 from the formula for the geometric sequence for finite terms
[tex]S_n=\frac{a_1*(1-r^n)}{1-r}\\ \\ =\frac{\frac{2}{3}\cdot \left(1-\left(-\frac{1}{5}\right)^4\right)}{1-\left(-\frac{1}{5}\right)}\\ \\ \frac{\frac{2}{3}\left(1-\left(-\frac{1}{5}\right)^4\right)}{1+\frac{1}{5}}=\frac{\frac{416}{625}}{\frac{6}{5}}=\frac{416\cdot \:5}{625\cdot \:6}=\frac{208}{375}[/tex]
Read more about geometric sequences here:
https://brainly.com/question/26147596
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