Answer:
The series has 7 terms
[tex]\displaystyle S_7=\frac{4372}{243}[/tex]
Step-by-step explanation:
Geometric Series
In the geometric series, each term is found by multiplying (or dividing) the previous term by a fixed number, called the common ratio.
We are given the series:
-12, -4, -4/3, ..., -4/243
We can find the common ratio by dividing one term by the previous term:
[tex]\displaystyle r=\frac{-4}{-12}[/tex]
Simplifying:
[tex]\displaystyle r=\frac{1}{3}[/tex]
Sum of terms: Given a geometric series with first term a1 and common ratio r, the sum of n terms is:
[tex]\displaystyle S_n=a_1\frac{1-r^n}{1-r}[/tex]
We need to find how many terms the series has. Using the explicit formula of a geometric series:
[tex]a_n=a_1\cdot r^{n-1}[/tex]
The last term is an=-4/243 and the first term is a1=-12. Solving for n:
[tex]\displaystyle n= \frac{\log(a_n/a_1)}{\log r}+1[/tex]
[tex]\displaystyle \frac{\log(-4/243/-12)}{\log 1/3}+1[/tex]
[tex]\displaystyle \frac{\log(1/729)}{\log 1/3}+1[/tex]
n=7
The series has 7 terms
Thus, the sum of the 7 terms of the series is:
[tex]\displaystyle S_7=-12\frac{1-(1/3)^7}{1-(1/3)}[/tex]
[tex]\mathbf{\displaystyle S_7=\frac{4372}{243}}[/tex]
