Answer:
The series has 7 terms
![\displaystyle S_7=\frac{4372}{243}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20S_7%3D%5Cfrac%7B4372%7D%7B243%7D)
Step-by-step explanation:
<u>Geometric Series</u>
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:
![\displaystyle r=\frac{-4}{-12}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20r%3D%5Cfrac%7B-4%7D%7B-12%7D)
Simplifying:
![\displaystyle r=\frac{1}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20r%3D%5Cfrac%7B1%7D%7B3%7D)
Sum of terms: Given a geometric series with first term a1 and common ratio r, the sum of n terms is:
![\displaystyle S_n=a_1\frac{1-r^n}{1-r}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20S_n%3Da_1%5Cfrac%7B1-r%5En%7D%7B1-r%7D)
We need to find how many terms the series has. Using the explicit formula of a geometric series:
![a_n=a_1\cdot r^{n-1}](https://tex.z-dn.net/?f=a_n%3Da_1%5Ccdot%20r%5E%7Bn-1%7D)
The last term is an=-4/243 and the first term is a1=-12. Solving for n:
![\displaystyle n= \frac{\log(a_n/a_1)}{\log r}+1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20n%3D%20%5Cfrac%7B%5Clog%28a_n%2Fa_1%29%7D%7B%5Clog%20r%7D%2B1)
![\displaystyle \frac{\log(-4/243/-12)}{\log 1/3}+1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Clog%28-4%2F243%2F-12%29%7D%7B%5Clog%201%2F3%7D%2B1)
![\displaystyle \frac{\log(1/729)}{\log 1/3}+1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Clog%281%2F729%29%7D%7B%5Clog%201%2F3%7D%2B1)
n=7
The series has 7 terms
Thus, the sum of the 7 terms of the series is:
![\displaystyle S_7=-12\frac{1-(1/3)^7}{1-(1/3)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20S_7%3D-12%5Cfrac%7B1-%281%2F3%29%5E7%7D%7B1-%281%2F3%29%7D)
![\mathbf{\displaystyle S_7=\frac{4372}{243}}](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cdisplaystyle%20S_7%3D%5Cfrac%7B4372%7D%7B243%7D%7D)