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