Chapter 4, Problem 30CR

To determine

To write: the nth term and find $a_{30}$ .

Answer to Problem 30CR

The nth term of the sequence is $a_n=3n-12;$

The $30th$ term of the arithmetic sequence is $78$

Explanation of Solution

Given:

  $-9,-6,-3,0,\mathrm{...}$

Formula Used:

  $a_n=a_1+\left(n-1\right)d$

Calculation:

Let $a_n$ , be the $nth$ term of an arithmetic sequence with first term $a_1$ , and common difference $d$ . The $nth$ term is given by:

  $a_n=a_1+\left(n-1\right)d$

Step 1 Consider two consecutive termsof the arithmetic sequence to find the common difference, $a_1=-9and{a}_2\text{=}-6$

  $\begin{array}{l}\begin{array}{l}Commondifference,\\ d=a_2-a_1\end{array}\hfill \\ \begin{array}{l}=-6-\left(-9\right)\\ =-6+9\\ =3\end{array}\hfill \end{array}$

The common difference of the given arithmetic sequence is $3$ .

Step 2

Write the equation for an arithmetic sequence.

  $a_n=a_1+\left(n-1\right)d$

Substitute $-9$ for $a_1$ and $3$ for $d$ .

  $a_n=-9+\left(n-1\right)\left(3\right)$

Distribute multiplication by $3$ .

  $a_n=-9+\left(3n\right)-\left(3\right)$

  $a_n=-9-3+3n$

Simplify.

  $a_n=3n-12$

Step 3 Use the equation to find the $30th$ term.

Write the equation.

  $a_n=3n-12$

Substitute $30forn$ .

  $\begin{array}{l}{a}_{30}=3\left(30\right)-12\\ =90-12\end{array}$

Simplify.

  $=78$

  $\Rightarrow$ The $30th$ term of the arithmetic sequence is $78$ .

Conclusion:

The nth term of the sequence is $a_n=3n-12;$

The $30th$ term of the arithmetic sequence is $78$