Chapter 8.5, Problem 23E

To determine

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

Answer to Problem 23E

  $a_n=a_{n-1}+1$ .

Explanation of Solution

Given:

The given graph is shown below:

  

Calculation:

According to the given graph, the sequence is $1,2,3,4,\mathrm{......}$ .

Therefore, the terms of the given sequence are $a_1=1,a_2=2,a_3=3,\mathrm{.....}\text{ and so on}$ .

Now, from the given terms, it is found that

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

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

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

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