Chapter 7.5, Problem 23E

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 work is:

Beginning with the second term in the sequence, each term $a_n$ equals $a_{n-1}-3$ . So a recursive rule is given by: $a_n=a_{n-1}-3$

The correct recursive rule is $a_n=a_{n-2}-a_{n-1}$ .

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: $a_n=a_{n-1}-3$

Now check if the given recursive rule is correct: $a_n=a_{n-1}-3$ .

  $\begin{array}{c}{a}_2=a_{2-1}-3\\ =a_1-3\\ =5-3\\ =2\end{array}$

Now find $a_3$ ,

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

Now find $a_4$ ,

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

It can be seen that the resultant third and fourth terms using the given recursive rule are not equal as given in the sequence. Thus the given recursive rule is wrong.

Now consider the sequence $5,2,3,-1,4,\cdots$ ,

It can be seen that the third term can be obtained by subtracting the second term from the first term.

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

Similarly, the fourth term can be obtained by subtracting the third term from the second term.

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

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