Question

Pleasee help mee ;-; i can't stand algebra, determine how many terms the geometric series has, and then find the sum of the series
-12 -4 - 4/3 - . . . - 4/243

Answer 1

Answer:

The series has 7 terms

$\displaystyle S_7=\frac{4372}{243}$

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:

$\displaystyle r=\frac{-4}{-12}$

Simplifying:

$\displaystyle r=\frac{1}{3}$

Sum of terms: Given a geometric series with first term a1 and common ratio r, the sum of n terms is:

$\displaystyle S_n=a_1\frac{1-r^n}{1-r}$

We need to find how many terms the series has. Using the explicit formula of a geometric series:

$a_n=a_1\cdot r^{n-1}$

The last term is an=-4/243 and the first term is a1=-12. Solving for n:

$\displaystyle n= \frac{\log(a_n/a_1)}{\log r}+1$

$\displaystyle \frac{\log(-4/243/-12)}{\log 1/3}+1$

$\displaystyle \frac{\log(1/729)}{\log 1/3}+1$

n=7

The series has 7 terms

Thus, the sum of the 7 terms of the series is:

$\displaystyle S_7=-12\frac{1-(1/3)^7}{1-(1/3)}$

$\mathbf{\displaystyle S_7=\frac{4372}{243}}$