Chapter 9.3, Problem 23E

Solution

a.

To calculate: The common difference of the sequence $-5,-2,1,4,\mathrm{....}$

The common difference of the sequence is $3$ .

Given information:

The sequence is $-5,-2,1,4,\mathrm{....}$

Formula used:

The common difference ( d ) is $d=a_2-a_1$ where $a_2$ is the second term of the sequence and $a_1$ is the first term of the sequence. Or common difference is the difference between two consecutive terms.

Calculation:

Consider the given sequence.

  $-5,-2,1,4,\mathrm{....}$

Use the common difference formula.

  $\begin{array}{c}d=-2-\left(-5\right)\\ =3\end{array}$

The common difference is $3$ .

b.

To calculate: The tenth term of the sequence $-5,-2,1,4,\mathrm{....}$

The tenth term of the given sequence is $22$ .

Given information:

The given sequence is $-5,-2,1,4,\mathrm{....}$

Formula used:

The formula for the nth term of an arithmetic sequence is $a_n=a+\left(n-1\right)d$ where a is the first term n is the nth term and d is the common difference.

Calculation:

Consider the given sequence.

The common difference is 3, the first term is $\left(-5\right)$ and the value of n is 10.

Substitute the respective values into the formula $a_n=a+\left(n-1\right)d$ .

  $\begin{array}{c}{a}_{10}=\left(-5\right)+\left(10-1\right)3\\ =-5+\left(9\right)3\\ =22\end{array}$

The tenth term of the given sequence is 22.

c.

To identify: The recursive rule for the nth term of the sequence $-5,-2,1,4,\mathrm{....}$

The recursive rule is $a_1=\left(-5\right),a_n=a_{n-1}+3\text{ for }n\ge 2$ .

Given information:

The given sequence is $-5,-2,1,4,\mathrm{....}$

Explanation:

Consider the given sequence.

Let the first term of the sequence is $a_1$ thus the formula for the nth term will be $a_n=a_{n-1}+d\text{ for }n\ge 2$ where d is the common difference.

Since the first term is $-5$ and the common difference is 3 thus,

  $a_1=\left(-5\right),a_n=a_{n-1}+3\text{ for }n\ge 2$

d.

To identify: An explicit rule for the nth term of the sequence $-5,-2,1,4,\mathrm{....}$

The explicit rule is $a_n=3n-8$ .

Given information:

The given sequence is $-5,-2,1,4,\mathrm{....}$

Explanation:

Consider the given sequence.

It is known that the common difference is 3, and the first term is $\left(-5\right)$ .

Substitute the respective values into the formula $a_n=a+\left(n-1\right)d$ .

  $\begin{array}{c}{a}_n=\left(-5\right)+\left(n-1\right)3\\ =-5+3n-3\\ =3n-8\end{array}$

The explicit rule is $a_n=3n-8$ .