Chapter 8.2, Problem 10E

To determine

To identify: Whether the sequence $4,9,14,19,24,\mathrm{...}$ is arithmetic or not.

Answer to Problem 10E

The sequence $4,9,14,19,24,\mathrm{...}$ is arithmetic because the difference between each two term is common.

Explanation of Solution

Given:

The sequence $4,9,14,19,24,\mathrm{...}$

Concept used:

A sequence is arithmetic when the differences between consecutive terms are the same. So, the sequence

  $a_1,a_2,a_3,\mathrm{....},a_n,\mathrm{.....}$ is arithmetic when there is a number $d$ such that

  $a_2-a_1=a_3-a_2=a_4-a_3=a_{n+1}-a_n=\mathrm{.....}=d$

The number $d$ is called the common difference of the arithmetic sequence.

Calculation:

Calculate the difference between $a_2$ and $a_1$ .

  $\begin{array}{c}{d}_1=a_2-a_1\\ =9-4\\ =5\end{array}$

Calculate the difference between $a_3$ and $a_2$ .

  $\begin{array}{c}{d}_2=a_3-a_2\\ =14-9\\ =5\end{array}$

Calculate the difference between $a_4$ and $a_3$ .

  $\begin{array}{c}{d}_3=a_4-a_3\\ =19-14\\ =5\end{array}$

Calculate the difference between $a_5$ and $a_4$ .

  $\begin{array}{c}{d}_4=a_5-a_4\\ =24-19\\ =5\end{array}$

From the above calculation the difference between each two term is common. So the sequence is arithmetic.