Chapter 6.7, Problem 41E

To determine

To write a recursive rule for the sequence $2,4,2,-2,-4,-2,\mathrm{......}$ . And write next two terms of the sequence .

Answer to Problem 41E

The recursive rule is $a_1=2,a_2=4,a_n=a_{n-1}-a_{n-2}$ and the next two terms of the given sequence are $2,4.$

Explanation of Solution

Given information :

Given sequence $2,4,2,-2,-4,-2,\mathrm{......}$ .

The given $2,4,2,-2,-4,-2,\mathrm{......}$ is not arithmetic sequence because there does not exists common difference

  $\begin{array}{l}d=a_2-a_1=4-2=2andalso{a}_3-a_2=2-4=-2\\ \Rightarrow 2\ne -2\end{array}$ .

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

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{4}{2}=2andalso\frac{a_3}{a_2}=\frac{2}{4}=\frac{1}{2}\\ \Rightarrow 2\ne \frac{1}{2}\end{array}$

Thus, now find $a_2-a_1=4-2=2$ is the third term ( $a_3$ )

Similar way, $a_3-a_2=2-4=-2$ is the fourth term ( $a_4$ )

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

Therefore, a recursive equation for the given sequence is $a_1=2,a_2=4,a_n=a_{n-1}-a_{n-2}$

Next two term of the given sequence is $2,4.$

Hence, the recursive rule is $a_1=2,a_2=4,a_n=a_{n-1}-a_{n-2}$ and the next two terms of the given sequence are $2,4.$