Chapter 6.6, Problem 42E

To determine

To write a recursive rule for the sequence $6,1,7,8,15,23,\mathrm{......}$ . And write next two terms of the sequence .

Answer to Problem 42E

The recursive rule is $a_1=6,a_2=1,a_n=a_{n-1}+a_{n-2}$ and next two terms of the given sequence are $38,66.$

Explanation of Solution

Given information:

Given sequence $6,1,7,8,15,23,\mathrm{......}$ .

The given $6,1,7,8,15,23,\mathrm{......}$ is not arithmetic sequence because there does not exists common difference

  $\begin{array}{l}d=a_2-a_1=1-6=-5andalso{a}_3-a_2=7-1=6\\ \Rightarrow -5\ne 6\end{array}$ .

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

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{1}{6}andalso\frac{a_3}{a_2}=\frac{7}{1}\\ \Rightarrow \frac{1}{6}\ne 7\end{array}$

Thus, now find $a_1+a_2=6+1=7$ is the third term ( $a_3$ )

Similar way, $a_2+a_3=1+7=8$ is the fourth term ( $a_4$ )

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

Therefore, a recursive equation for the given sequence is $a_1=6,a_2=1,a_n=a_{n-1}+a_{n-2}$

Next two term of the given sequence are $38,66.$

Hence, recursive rule is $a_1=6,a_2=1,a_n=a_{n-1}+a_{n-2}$ and next two terms of the given sequence are $38,66.$