Chapter 7.5, Problem 22E

Solution

To state: The error and correct the error in writing a recursive rule for the sequence $5,2,3,-1,4,\cdots$ .

The student’s work is:

Beginning with the third term in the sequence, each term $a_n$ equals $a_{n-2}-a_{n-1}$ . So a recursive rule is given by: $a_n=a_{n-2}-a_{n-1}$

The recursive rule is correct.

Given information:

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

Explanation:

Consider the given sequence,

  $5,2,3,-1,4,\cdots$

The given recursive rule is: $a_n=a_{n-2}-a_{n-1}$

Now check if the given recursive rule is correct: $a_n=a_{n-2}-a_{n-1}$ .

  $\begin{array}{c}{a}_3=a_{3-2}-a_{3-1}\\ =a_1-a_2\\ =5-2\\ =3\end{array}$

Now find $a_4$ ,

  $\begin{array}{c}{a}_4=a_{4-2}-a_{4-1}\\ =a_2-a_3\\ =2-3\\ =-1\end{array}$

Now find $a_5$ ,

  $\begin{array}{c}{a}_5=a_{5-2}-a_{5-1}\\ =a_3-a_4\\ =3-\left(-1\right)\\ =4\end{array}$

It can be seen that the given recursive rule is correct. There is no error in the recursive rule.

So the recursive rule for the sequence $a_0=5,a_1=2$ is $a_n=a_{n-2}-a_{n-1}$ .