Chapter 9.1, Problem 22E

To determine

To Find: The parameters of a geometric sequence, two of whose terms are given.

Answer to Problem 22E

The first term is $\frac{1}{4}$ , the common ratio is $-2$ and the explicit rule for the $n^{th}$ term is $a_n=\frac{1}{4}{\left(-2\right)}^{n-1}$ .

Explanation of Solution

Given Information: The $2^{nd}$ term and the $7^{th}$ term of the geometric sequence are $-\frac{1}{2}$ and $16$ respectively.

Concept Used:

• The $n^{th}$ term of a geometric series with first term $a_1$ and common ratio $r$ is given by
$a_n=a_1{r}^{n-1}$

Calculation:The second term is $a_2=-\frac{1}{2}$ . So

  $a_2=a_1{r}^{2-1}=-\frac{1}{2}$

  $\Rightarrow a_{1}r=-\frac{1}{2}$ (1)

The seventh term is $16$ . So

  $a_7=a_1{r}^{7-1}=16$

  $\Rightarrow a_1{r}^6=16$ (2)

Now, divide equation (1) by equation (2)to write

  $\frac{a_1{r}^6}{a_{1}r}=\frac{16}{\left(-\frac{1}{2}\right)}$

  $\Rightarrow r^5=-32$

  $\Rightarrow r=-2$

Put the calculated value $r=-2$ in equation (1) to write

  $a_1\times (-2)=-\frac{1}{2}$

  $\Rightarrow a_1=\frac{1}{4}$

Therefore, the first term is $\frac{1}{4}$ , the common ratio is $-2$ and the explicit rule for the $n^{th}$ term is $a_n=\frac{1}{4}{\left(-2\right)}^{n-1}$ .