Question

Tell weather each sequence is arithmetic sequence or not. Shortly explain why or why not for each sequence!

Answer 1

Given the sequence:

Sequence 1:

$\frac{1}{2},\frac{7}{6},\frac{11}{6},\frac{5}{2}$

Sequence 2:

$\frac{1}{2},\frac{1}{3},\frac{2}{9},\frac{4}{27}$

Let's tell whether each sequence is an arithmetic sequence.

An Arithmetic sequence will have a common difference.

The difference between the subsequent terms will be the same.

Let's subtract each term from the next term.

• Sequence 1:

$\begin{gathered} \frac{5}{2}-\frac{11}{6}=\frac{3(5)-1(11)}{6}=\frac{15-11}{6}=\frac{4}{6}=\frac{2}{3} \\ \\ \frac{11}{6}-\frac{7}{6}=\frac{11-7}{6}=\frac{4}{6}=\frac{2}{3} \\ \\ \frac{7}{6}-\frac{1}{2}=\frac{1(7)-3(1)}{6}=\frac{7-3}{6}=\frac{4}{6}=\frac{2}{3} \end{gathered}$

Sequence 1 has a common difference of 2/3.

Since the difference is common, sequence 1 is an arithmetic sequence.

• Sequence 2:

$\begin{gathered} \frac{4}{27}-\frac{2}{9}=\frac{1(4)-3(2)}{27}=\frac{4-6}{27}=-\frac{2}{27} \\ \\ \frac{2}{9}-\frac{1}{3}=\frac{1(2)-3(1)}{9}=\frac{2-3}{9}=-\frac{1}{9} \\ \\ \frac{1}{3}-\frac{1}{2}=\frac{2-3}{6}=-\frac{1}{6} \end{gathered}$

Sequence 2 does not have a common difference.

Therefore, sequence 2 is NOT an arithmetic sequence.

ANSWER:

• Sequence 1 is an arithmetic sequence.

,

• Sequence 2 is NOT an arithmetic sequence.