Question

Determine if each sequence is arithmetic. If so, indicate the common difference.

$\begin{array}{l}\left(a\right)9,20,31,42,53,64,\dots \\ \left(b\right)12,6,0,-6,-12,-18,\dots \\ \left(c\right)7,1,10,4,13,7,\dots \end{array}$

Short answer

Part (a) The sequence is arithmetic.

Part (b) The sequence is arithmetic.

Part (c) The sequence is not arithmetic.

Step-by-step solution

Part (a) Step 1. Find the consecutive difference of the sequence.

Consider the given sequence: $\left(a\right)9,20,31,42,53,64,\dots$

Find the difference between the consecutive terms.

$\begin{array}{rcl}20-9& =& 11\\ 31-20& =& 11\\ 42-31& =& 11\\ 53-42& =& 11\\ 64-53& =& 11\end{array}$

• Since the difference of all consecutive terms is the same.
• The common difference is d= 11.
• The sequence is an arithmetic sequence.

Part (b) Step 1. Find the consecutive difference of the sequence.

Consider the given sequence:$\left(b\right)12,6,0,-6,-12,-18$

Find the difference between the consecutive terms.

$\begin{array}{rcl}6-12& =& -6\\ 0-6& =& -6\\ -6-0& =& -6\\ -12-\left(-6\right)& =& -6\\ -18-\left(-12\right)& =& -6\end{array}$

• Since the difference of all consecutive terms is the same.
• The common difference is d= -6.
• The sequence is an arithmetic sequence.

Part (c) Step 1. Find the consecutive difference of the sequence.

Consider the given sequence: $\left(c\right)7,1,10,4,13,7,\dots$

Find the difference between the consecutive terms.

$\begin{array}{rcl}1-7& =& -6\\ 10-1& =& 9\\ 4-10& =& -6\\ 13-4& =& 9\\ 7-13& =& -6\end{array}$

• Since the difference of all consecutive terms is not the same.
• The sequence is not an arithmetic sequence.