Chapter 8.2, Problem 11E

a.

To determine

To calculate: The arithmetic sequence.

Answer to Problem 11E

The $nth$ rule is $a_n=6n-9$ and sequence is $-3,3,9,\mathrm{.....}$ .

Explanation of Solution

Given information:

The first term is $-3$ and each term is 6 less than the previous term.

Formula used:

Arithmetic difference − the difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d .

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

According to the question ,

  $\begin{array}{l}{a}_1=-3\\ d=6\end{array}$

Therefore, the $nth$ term is

  $\begin{array}{l}=a_n=a_1+(n-1)d\\ =a_n=(-3)+(n-1)6\\ =-3+6n-6\\ =6n-9\end{array}$

The arithmetic sequence is $-3,3,9,\mathrm{.....}$

Hence, the $nth$ rule is $a_n=6n-9$ and sequence is $-3,3,9,\mathrm{.....}$ .

b.

To determine

To calculate: The arithmetic sequence.

Answer to Problem 11E

The $nth$ rule is $a_n=6n-9$ and sequence is $-3,3,9,\mathrm{.....}$ .

Explanation of Solution

Given information:

The first term is 7 and each term is 5 more than the previous term.

Formula used:

Odd numbers − are whole numbers that cannot be divided exactly into two pairs.

The difference of consecutive term is constant. The constant difference is called the common difference and is denoted by d .

Therefore,

  $d=a_{n+1}-a_n;n\ge 1$ .

The $nth$ tem of an arithmetic sequence with first term $a_1$ and common difference $d$ is given by : $a_n=a_1+(n-1)d$ .

Calculation:

According to the question ,

  $\begin{array}{l}{a}_1=7\\ d=5\end{array}$

Therefore,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_n=7+(n-1)5\\ =7+5n-5\\ =5n+2\end{array}$

The sequence

  $\begin{array}{l}{a}_2=5(2)+2\\ =10+2\\ =12\\ a_3=5(3)+2\\ =15+2\\ =17\end{array}$

And so on.

The sequence is $7,12,17,\mathrm{........}$

Hence, the $nth$ rule is $a_n=5n+2$ and sequence is $7,12,17,\mathrm{.....}$ .