Chapter 7.5, Problem 18E

Solution

To state: The recursive rule for the sequence $1,4,5,9,14,\cdots$ .

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

Given information:

The given sequence is $1,4,5,9,14,\cdots$ .

Explanation:

Consider the given sequence,

  $1,4,5,9,14,\cdots$

The first term of the sequence is: $a_1=1$

The common difference ( d ) is:

  $\begin{array}{c}4-1=3\\ 5-4=1\end{array}$

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:

  $\begin{array}{c}r=\frac{4}{1}\\ =4\\ r=\frac{5}{4}\\ =1.25\end{array}$

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

Now again consider the sequence $1,4,5,9,14,\cdots$ ,

Beginning with the third term in the sequence, each term is the sum of the two previous terms.

So the recursive rule for the sequence $a_0=1,a_1=4$ is $a_n=a_{n-2}+a_{n-1}$ .