Chapter 8.5, Problem 28E

To determine

To describe and correct: the error for writing a recursive rule for the given sequence.

Answer to Problem 28E

Error, $a_n=a_{n-2}-a_{n-1}$ .

Explanation of Solution

Given:

The given sequence is $5,2,3,-1,-4$ .

Beginning with the second term in the sequence, and recursive rule is given by $a_n=a_{n-2}-3$ .

Calculation:

The recursive rule is given by $a_n=a_{n-2}-3$ .

And the given sequence is $5,2,3,-1,-4$ .

Now, substitutes the value of $n=1,2,3,\mathrm{.....}\text{ and so on}$ , so that to check the error. Therefore,

  $\begin{array}{l}{a}_n=a_{n-2}-3\{\text{given sequences}}\\ a_3=a_{3-2}-3\{\text{substitues }n=1\text{}}\\ a_3=a_1-3\\ \Rightarrow 3=5-3\{\text{false}}\end{array}$

Thus, the error in the formula for the recursive rule. The required formula is $a_n=a_{n-2}-a_{n-1}$ . Therefore,

  $\begin{array}{l}{a}_n=a_{n-2}-a_{n-1}\\ a_3=a_{3-2}-a_{3-1}\{\text{substitues }n=1\text{}}\\ a_3=a_1-a_2\\ \Rightarrow 3=5-2\{\text{true}}\end{array}$

Hence, the error in recursive rule for the given sequence is $a_n=a_{n-2}-3$ , and the correct formula is

  $a_n=a_{n-2}-a_{n-1}$ .