Chapter 6.6, Problem 57E

To determine

Tocomplete: The given sentence by filling in the blank.

Answer to Problem 57E

Complete sentence: A recursive rule gives the beginning term (s) of a sequence and $a_n=f\left(a_{n-1}\right)$ that tells how $a_n$ is related to one or more preceding terms.

Explanation of Solution

Given information:

The partial sentence is: A recursive rule gives the beginning term (s) of a sequence and $a_n$ ----, that tells how $a_n$ is related to one or more preceding terms.

A recursive rule/formula is written in two parts: (1) the statement of the first term, and (2) the statement of the formula relating to the term $a_n$ and its preceding term $a_{n-1}$ .

The relation can be written as: First term $a_0=\text{a}\text{number}$ ; $a_n=f\left(a_{n-1}\right)$ .

Example: The rule: $a_0=\text{1}$ ; $a_n=a_{n-1}+3$ forms the sequence $\langle a_n\rangle =a_0,a_1,a_2\dots$ whose terms are $a_1=a_0+3=1+3=4$ , $a_2=a_1+3=4+3=7$ , $a_3=a_2+3=7+3=11$ so the sequence is $\langle a_n\rangle =1,4,7,11,\dots$

Hence, the complete sentence is:

The partial sentence is: A recursive rule gives the beginning term (s) of a sequence and $a_n=f\left(a_{n-1}\right)$ that tells how $a_n$ is related to one or more preceding terms.

Conclusion:

Complete sentence:A recursive rule gives the beginning term (s) of a sequence and $a_n=f\left(a_{n-1}\right)$ that tells how $a_n$ is related to one or more preceding terms.