Chapter 11.3, Problem 25PPS

To determine

Write an equation for the nth term of the geometric series $-1,1,-1,\mathrm{.......}$

Answer to Problem 25PPS

  $a_n={(-1)}^n$

Explanation of Solution

Given: The series $-1,1,-1,\mathrm{.......}$

Concept Used:

General equation for the nth term of any geometric series is $a_n=a_1·r^{n-1}$

Where $a_1$ the first term and r is the common ratio

Calculation:

  $a_1=-1;\text{Common}\text{ratio}=r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{1}{-1}=-1$

  $\begin{array}{l}{a}_1=-1,r=-1,\text{Find}{a}_n\\ $ a_n=a_1·r^{n-1}=-1·{(-1)}^{n-1}={(-1)}^1·{(-1)}^{n-1}={(-1)}^{1+n-1}={(-1)}^n$$ a_n={(-1)}^n$\end{array}$

Thus, nth term of the series $-1,1,-1,\mathrm{.......}$ is $a_n={(-1)}^n$