Chapter 6.6, Problem 43E

To determine

To write a recursive rule from the given graph of a sequence and next two terms

Answer to Problem 43E

The recursive rule is $a_1=1,a_2=3,a_n=\left(a_{n-1}\right)\left(a_{n-2}\right)$ and next two terms of the given sequence are $243,6561.$

Explanation of Solution

Given information:

Given graph of given sequence

  

The sequence $1,3,3,9,27,\mathrm{......}$ obtained from given graph is not arithmetic sequence because there does not exists common difference

  $\begin{array}{l}d=a_2-a_1=3-1=2andalso{a}_3-a_2=3-3=0\\ \Rightarrow 2\ne 0\end{array}$ .

Similar way the given sequence is not a geometric sequence as well because there does not common ratio

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{3}{1}=3andalso\frac{a_3}{a_2}=\frac{3}{3}=1\\ \Rightarrow 3\ne 1\end{array}$

Thus, now find $a_1\times a_2=1\times 3=3$ is the third term ( $a_3$ )

Similar way, $a_2\times a_3=3\times 3=9$ is the fourth term ( $a_4$ )

Thus, each term is the product of the two previous terms,

Therefore, a recursive equation for the given sequence is $a_1=1,a_2=3,a_n=\left(a_{n-1}\right)\left(a_{n-2}\right)$

Next two term of the given sequence are $243,6561.$

Hence, recursive rule is $a_1=1,a_2=3,a_n=\left(a_{n-1}\right)\left(a_{n-2}\right)$ and next two terms of the given sequence are $243,6561.$