Chapter 7.8, Problem 43PPE

To determine

To find common ratio and write explicit and a recursive formulas, the given $200,-100,50,-25,\mathrm{.......}$ .

Answer to Problem 43PPE

The common ratio $=-\frac{1}{2}$ and the explicit rule is $a_n=200{\left(-\frac{1}{2}\right)}^{n-1}$ and the recursive rule is $a_n=\left(-\frac{1}{2}\right)a_{n-1},a_1=200,\text{where}n=2,3,4,5,\mathrm{.......}$

Explanation of Solution

Given information :

Given geometric sequence $200,-100,50,-25,\mathrm{.......}$

The given sequence $200,-100,50,-25,\mathrm{.......}$ is a geometric sequence because there exists an common ratio

  $\begin{array}{l}r=\frac{a_2}{a_1}=-\frac{100}{200}=-\frac{1}{2}\text{and}\text{also}\frac{a_3}{a_2}=\frac{50}{-100}=-\frac{1}{2}\\ \Rightarrow -\frac{1}{2}=-\frac{1}{2}\end{array}$ .

And $a_1=200$

Thus, the explicit rule or $n^{th}\text{term}$ for a geometric sequence is $a_n=a_1{r}^{n-1}$

  $\therefore a_n=200{\left(-\frac{1}{2}\right)}^{n-1}$

Thus, recursive rule for geometric sequence is $a_n=r{a}_{n-1}$

  $\Rightarrow a_n=\left(-\frac{1}{2}\right)a_{n-1},a_1=200,\text{where}n=2,3,4,5,\mathrm{.......}$

Hence, the common ratio $=-\frac{1}{2}$ and the explicit rule is $a_n=200{\left(-\frac{1}{2}\right)}^{n-1}$ and the recursive rule is $a_n=\left(-\frac{1}{2}\right)a_{n-1},a_1=200,\text{where}n=2,3,4,5,\mathrm{.......}$