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:

`(1)/(2),(7)/(6),(11)/(6),(5)/(2)`

Sequence 2:

`(1)/(2),(1)/(3),(2)/(9),(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} (5)/(2)-(11)/(6)=(3(5)-1(11))/(6)=(15-11)/(6)=(4)/(6)=(2)/(3) \\ \\ (11)/(6)-(7)/(6)=(11-7)/(6)=(4)/(6)=(2)/(3) \\ \\ (7)/(6)-(1)/(2)=(1(7)-3(1))/(6)=(7-3)/(6)=(4)/(6)=(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} (4)/(27)-(2)/(9)=(1(4)-3(2))/(27)=(4-6)/(27)=-(2)/(27) \\ \\ (2)/(9)-(1)/(3)=(1(2)-3(1))/(9)=(2-3)/(9)=-(1)/(9) \\ \\ (1)/(3)-(1)/(2)=(2-3)/(6)=-(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.