Question

Which of the following is an arithmetic sequence?

2, 4, 8, 16,...
12, 4, 4/3, 16/3...
1/2, -1/2, -3/2, -5/2 ...
1/2, -1/2, 1/2, -1/2 ...

Answer 1

Answer: Number 3. (1/2, -1/2, -3/2, -5/2 ...)

Explanation:

1/2 - 1 = -1/2. -1/2 - 1 = -3/2. Etc.

Answer 2

The third sequence.

Explanation:

In an arithmetic sequence, the difference between two consecutive terms is the same.

For each option, find the difference between consecutive terms:

First option:

• 
  `4 - 2 = 2`.
• 
  `8 - 4 = 4`.
• 
  `16 - 8 = 8`.

The differences are not the same. As a result, this option is not an arithmetic sequence.

Second option:

• 
  `4 - 12 = -8`.
• 
  `\displaystyle (4)/(3) - 4 = -(8)/(3)`.
• 
  `\displaystyle (16)/(3) - (4)/(3) = (12)/(3) = 4`.

The differences are not the same. As a result, this option is not an arithmetic sequence, either.

Third option:

• 
  `\displaystyle -(1)/(2) - (1)/(2) = -1`.
• 
  `\displaystyle -(3)/(2) - \left(-(1)/(2)\right) = -(3)/(2) + (1)/(2) = -1`.
• 
  `\displaystyle -(5)/(2) - \left(-(3)/(2)\right) = -(5)/(2) + (3)/(2) = -1`.

The differences are all
`-1`. As a result, this option is indeed an arithmetic sequence. Its common difference is
`(-1)`.

Fourth option:

• 
  `\displaystyle -(1)/(2) - (1)/(2) = -1`.
• 
  `\displaystyle (1)/(2) - \left(-(1)/(2)\right) = (1)/(2) + (1)/(2) = 1`.
• 
  `\displaystyle -(1)/(2)\right - (1)/(2) = -1`.

The differences are varying between
`1` and
`-1`. As a result, this option is not an arithmetic sequence.