Chapter 6.7, Problem 1E

To determine

Tocomplete: The given sentence by filling in the blank.

Answer to Problem 1E

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:

  $a_0=\text{36}$ ; $a_n=a_{n-1}-3$ forms the sequence $\langle a_n\rangle =a_0,a_1,a_2\dots$ whose terms are

  $a_1=a_0-3=36-3=33$ , $a_2=a_1-3=33-3=30$ , $a_3=a_2-3=30-3=27$ so the sequence is $\langle a_n\rangle =36,33,30,27,\dots$

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.