Chapter 7.5, Problem 21E

Solution

To state: The recursive rule for the sequence $16,9,7,2,5,\cdots$ .

The resultant recursive rule is $a_n=\left(a_{n-2}\right)-\left(a_{n-1}\right)$ .

Given information:

The given sequence is $16,9,7,2,5,\cdots$ .

Explanation:

Consider the given sequence,

  $16,9,7,2,5,\cdots$

The first term of the sequence is: $a_1=16$

The common difference ( d ) is:

  $\begin{array}{c}9-16=-7\\ 7-9=-2\end{array}$

The given sequence is not an arithmetic sequence because the common difference of two successive terms is not equal.

Now check if the sequence is geometric.

The common ratio ( r ) is:

  $\begin{array}{c}r=\frac{9}{16}\\ =0.562\\ r=\frac{7}{9}\\ =0.778\end{array}$

The given sequence is not a geometric sequence because the common ratio is not same from the two terms.

Now again consider the sequence $16,9,7,2,5,\cdots$ ,

Beginning with the third term in the sequence, it can be seen that each term is the difference of two previous terms.

The third term is the difference of first and second term. Similarly, the fourth term is the difference of second and third.

So the recursive rule for the sequence $a_1=16,a_1=9$ is $a_n=\left(a_{n-2}\right)-\left(a_{n-1}\right)$ .