Chapter 8, Problem 1CC

To determine

(a)

To explain:

The sequence and the way to denote the terms of a sequence.

Explanation of Solution

A sequence is a function $a$ whose domain is the set of natural numbers. The terms of the sequence are the function values.

The terms of the sequence can be expressed as,

$a_1,a_2,a_3,\dots ,a_n,\dots$

Here, the first term is $a_1$, second term is $a_2$ and the general term $a_n$ is called the nth term.

The sequence with $n\text{th}$ term can be written as $a_n$.

To determine

(b)

To find:

The formula for sequence of even numbers and formula for sequence of odd numbers.

Answer to Problem 1CC

The formula for sequence of even numbers and formula for sequence of odd numbers are $a_n=2n$ and $a_n=2n-1$ respectively.

Explanation of Solution

Calculation:

The sequence of even number is,

$2,4,6,8,\cdots$

The first term of the sequence is $2$, the second term of the sequence is $4$, the third term is $6$ and common difference is calculated as,

$\begin{array}{c}d=a_2-a_1\\ =4-2\\ =2\end{array}$

Substitute 2 for $a$ and $2$ for $d$ the formula of the sequence of even numbers can be expressed as,

$\begin{array}{c}{a}_n=2+\left(n-1\right)\left(2\right)\\ =2+2n-2\\ =2n\end{array}$

The sequence of odd number is,

$1,3,5,7,\cdots$

The first term of the sequence is $1$, the second term of the sequence is $3$, the third term is $5$ and common difference is calculated as,

$\begin{array}{c}d=a_2-a_1\\ =3-1\\ =2\end{array}$

Substitute 2 for $a$ and $2$ for $d$ the formula of the sequence of odd numbers can be expressed as,

$\begin{array}{c}{a}_n=1+\left(n-1\right)\left(2\right)\\ =1+2n-2\\ =2n-1\end{array}$

Conclusion:

Hence, the formula for sequence of even numbers and formula for sequence of odd numbers are $a_n=2n$ and $a_n=2n-1$ respectively.

To determine

(c)

To find:

The first three terms and the 10^thterm of the sequence given by $a_n=\frac{n}{n+1}$.

Answer to Problem 1CC

The first three terms of the sequence $a_n=\frac{n}{n+1}$ are $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$ and the 10^thterm is $\frac{10}{11}$.

Explanation of Solution

Given information:

The sequence is,

$a_n=\frac{n}{n+1}$   .............(1)

Calculation:

Substitute $1$ for n in equation (1) to obtain the first term.

$\begin{array}{c}{a}_1=\frac{1}{1+1}\\ =\frac{1}{2}\end{array}$

Substitute 2 for n in equation (1) to obtain the second term.

$\begin{array}{c}{a}_2=\frac{2}{2+1}\\ =\frac{2}{3}\end{array}$

Substitute 3 for n in equation (1) to obtain the third term.

$\begin{array}{c}{a}_3=\frac{3}{3+1}\\ =\frac{3}{4}\end{array}$

Substitute 10 for n in equation (1) to obtain the tenth term.

$\begin{array}{c}{a}_{10}=\frac{10}{10+1}\\ =\frac{10}{11}\end{array}$

Conclusion:

Hence, the first three terms of the sequence $a_n=\frac{n}{n+1}$ are $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$ and the 10^thterm is $\frac{10}{11}$.