Chapter 11.3, Problem 31WE

To determine

To calculate:

The following sequence is arithmetic, geometric or neither and the formula for $n^{th}$ term:

  $200,-100,50,-25,\mathrm{...}$

Answer to Problem 31WE

The sequence is geometric sequence and formula for $n^{th}$ term is $a_n=200{\left(-\frac{1}{2}\right)}^{\left(n-1\right)}$

Explanation of Solution

Given information:

  $200,-100,50,-25,\mathrm{...}$

Calculation:

Check the common difference first to categorize the given sequence into arithmetic, geometric or neither:

  $\begin{array}{l}{a}_1=200\\ a_2=-100\\ a_3=50\\ a_4=-25\end{array}$

Now, the common difference is calculated as:

  $\begin{array}{l}d=a_2-a_1=-100-200=-300\\ d=a_3-a_2=-300+50=-250\\ d=a_4-a_3=-25+50=-75\end{array}$

Therefore, the given sequence is not arithmetic sequence as the common ratio is not same.

Now, check for geometric sequence:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{-100}{200}=-\frac{1}{2}\\ r=\frac{a_3}{a_2}=\frac{50}{-100}=-\frac{1}{2}\\ r=\frac{a_4}{a_3}=\frac{-25}{50}=-\frac{1}{2}\end{array}$

Therefore, the given sequence is geometric sequence.

Now, the formula for $n^{th}$ term is calculated as:

  $\begin{array}{l}{a}_n=a{r}^{\left(n-1\right)}\\ a_n=200{\left(-\frac{1}{2}\right)}^{\left(n-1\right)}\end{array}$