Chapter 8.2, Problem 2E

a.

To determine

To calculate: The sequence consists of all the positive odd numbers.

Answer to Problem 2E

The sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$ .

Explanation of Solution

Given information:

The positive odd numbers.

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 ,

The sequence of all positive odd numbers.

Let,

  $a_n=2n+1\text{ where }n=0,1,2,3,\mathrm{....}$

Therefore, the sequence is

  $1,3,5,7,9,11,13,15,\mathrm{........}$

With common difference 2.

Hence, the $nth$ rule is $a_n=2n+1.$

b.

To determine

To calculate: The sequence starts with 1 and has common difference of 2.

Answer to Problem 2E

The sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$

Explanation of Solution

Given information:

The sequence starts with 1 and has a common difference of 2.

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 ,

  $\begin{array}{l}{a}_1=1\\ d=2\end{array}$

Therefore,

  $\begin{array}{l}{a}_n=a_1+(n-1)d\\ a_2=a_1+(n-1)d\\ =1+\left(2-1\right)2\\ =1+4-2\\ =3\end{array}$

  $\begin{array}{l}{a}_3=a_1+(3-1)2\\ =1+\left(2\right)2\\ =1+4\\ =5\\ a_4=a_1+(4-1)2\\ =1+\left(3\right)2\\ =1+6\\ =7\end{array}$

And so on.

Hence, the sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$

c.

To determine

To calculate: The sequence that has $nth$ term of $a_n=1+\left(n-1\right)2$ .

Answer to Problem 2E

The sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$

Explanation of Solution

Given information:

The $nth$ term is $a_n=1+\left(n-1\right)2$ .

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 ,

  $a_n=1+\left(n-1\right)2$

Therefore,

  $a_1=1$

  $\begin{array}{l}{a}_2=1+(2-1)2\\ =1+\left(1\right)2\\ =1+2\\ =3\end{array}$

  $\begin{array}{l}{a}_3=1+(3-1)2\\ =1+\left(2\right)2\\ =1+4\\ =5\\ a_4=1+(4-1)2\\ =1+\left(3\right)2\\ =1+6\\ =7\end{array}$

And so on.

Hence, the sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$

d.

To determine

To calculate: The has an $nth$ term of $a_n=2n+1$ .

Answer to Problem 2E

The sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$

Explanation of Solution

Given information:

The $nth$ term $a_n=2n+1$

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 ,

  $nth$ term is $a_n=2n+1$

Therefore,

  $\begin{array}{l}{a}_n=2n+1\\ a_1=2(1)+1\\ =2+1\\ =3\end{array}$

  $\begin{array}{l}{a}_2=2(2)+1\\ =4+1\\ =5\\ a_3=2(3)+1\\ =6+1\\ =7\end{array}$

And so on.

Hence, the sequence is $1,3,5,7,9,11,13,15,\mathrm{........}$