Chapter 6.6, Problem 26E

To determine

To write an explicit rule for the recursive rule $a_1=-2,a_n=\left(9\right)a_{n-1}$ .

Answer to Problem 26E

The explicit rule for the given recursive rule is $a_n=\left(-2\right){\left(9\right)}^{n-1}$ .

Explanation of Solution

Given information:

Given recursive rule $a_1=-2,a_n=\left(9\right)a_{n-1}$ .

The given recursive rule $a_1=-2,a_n=\left(9\right)a_{n-1}$ represents a geometric sequence with first term $a_1=-2$ and common ratio $r=9$ .

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

  $\Rightarrow a_n=\left(-2\right){\left(9\right)}^{n-1}$

Hence, the explicit rule for the given recursive rule is $a_n=\left(-2\right){\left(9\right)}^{n-1}$ .