Chapter 8.5, Problem 26E

To determine

To write: a recursive rule for the sequence shown in the given graph.

Answer to Problem 26E

  $a_n=a_{n-1}-2$ .

Explanation of Solution

Given:

The given graph is shown below:

  

Calculation:

According to the given graph, the sequence is $4,2,0,-2,\mathrm{....}$ .

Therefore, the terms of the given sequence are $a_1=4,a_2=2,a_3=0,\mathrm{.....}\text{ and so on}$ .

Now, from the given terms, it is found that

  $\begin{array}{l}{a}_1=-2\{\text{given terms}}\\ a_n=a_{n-1}-2\end{array}$

Now, check this regressive rule for 2^nd and 3^rd term and so on.

  $\begin{array}{l}{a}_n=a_{n-1}-2\{\text{recursive rule}}\\ a_2=a_{2-1}-2\{\text{put }n=2\text{}}\\ a_2=a_1-2\\ a_2=4-2\{\text{given }{a}_1=4\text{}}\\ a_2=2\{\text{true}}\\ a_3=a_{3-1}-2\{\text{put }n=3\text{}}\\ a_3=a_2-2\\ a_3=2-2\{\text{given }{a}_2=2\text{}}\\ a_3=0\{\text{true}}\end{array}$

Hence, the recursive rule for the given sequence is $a_n=a_{n-1}-2$ .