Chapter 7.1, Problem 24E

Solution

To state: The pattern, the next term for the sequence $4.2,2.6,1,-0.6,-2.2,\dots$ .

The resultant term is $-3.8$ and rule is $a_n=5.8-1.6n$ .

Given information:

The given sequence is $4.2,2.6,1,-0.6,-2.2,\dots$ .

Explanation:

A sequence is a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that domain starts with 1.

Now consider the given sequence,

  $4.2,2.6,1,-0.6,-2.2,\dots$

It can be seen from the given sequence that the common difference of each term is $1.6$ . Then the first term can be written as: $5.8-1.6\left(1\right)=4.2$

Similarly, the given terms of the sequence can be written as:

  $5.8-1.6\left(1\right),5.8-1.6\left(2\right),5.8-1.6\left(3\right),5.8-1.6\left(4\right),5.8-1.6\left(5\right),\dots$

The next terms is:

  $\begin{array}{c}5.8-1.6\left(6\right)=5.8-9.6\\ =-3.8\end{array}$

A rule for the nth term is $a_n=5.8-1.6n$ .

Therefore, the next term of the given sequence is $-3.8$ and the rule it represents is $a_n=5.8-1.6n$ .