Chapter 8, Problem 28RE

To determine

Whether the given sequence is an arithmetic sequence. The common difference for an arithmetic sequence.

Answer to Problem 28RE

The given sequence is not an arithmetic sequence.

Explanation of Solution

Given information:

An sequence as $0,1,3,6,10,\mathrm{...}$ .

Concept used:

An arithmetic sequence of n terms, has the form $a_1,a_2,a_3,\mathrm{...},a_n$ and should have a common difference between each two consecutive terms.

That is

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

Common difference can be defined by d .

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

nth term of the arithmetic sequence has the form

  $a_n=a_1+\left(n-1\right)d$

Where $a_1$ is the first term of the sequence, and d is the common difference.

Calculation:

Now, consider the given sequence.

  $0,1,3,6,10,\mathrm{...}$

The terms of the sequence can be found as

  $\begin{array}{l}{a}_1=0\\ a_2=1\\ a_3=3\\ a_4=6\\ \mathrm{...}\end{array}$

Now, the difference can be calculated as shown:

  $\begin{array}{l}{d}_1=a_2-a_1=1-0=1\\ d_2=a_3-a_2=3-1=2\\ d_3=a_4-a_3=6-3=3\end{array}$

Now, the differences for each two consecutive terms are as

  $\begin{array}{l}{d}_1=1\\ d_2=2\\ d_3=3\end{array}$

Therefore, differences are not equal for each two consecutive terms of the sequence.

So, it is not an arithmetic sequence.