Chapter 6.6, Problem 40E

To determine

To write a recursive rule for the sequence $10,9,1,8,-7,15,\mathrm{......}$ . And write next two terms of the sequence .

Answer to Problem 40E

The recursive rule is $a_1=10,a_2=9,a_n=a_{n-2}-a_{n-1}$ and the next two terms of the given sequence are $-22,37$

Explanation of Solution

Given information:

Given sequence $10,9,1,8,-7,15,\mathrm{......}$ .

The given $10,9,1,8,-7,15,\mathrm{....}$ is not arithmetic sequence because there does not exists common difference

  $\begin{array}{l}d=a_2-a_1=9-10=-1andalso{a}_3-a_2=1-9=-8\\ \Rightarrow -1\ne -8\end{array}$ .

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

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{9}{10}andalso\frac{a_3}{a_2}=\frac{1}{9}\\ \Rightarrow \frac{9}{10}\ne \frac{1}{9}\end{array}$

Thus, now find $a_1-a_2=10-9=1$ is the third term ( $a_3$ )

Similar way, $a_2-a_3=9-1=8$ 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 $a_1=10,a_2=9,a_n=a_{n-2}-a_{n-1}$

Next two term of the given sequence are $-22,37.$

Hence, the recursive rule is $a_1=10,a_2=9,a_n=a_{n-2}-a_{n-1}$ and the next two terms of the given sequence are $-22,37$