Question

Which of the following is an arithmetic sequence?

Answer 1

Answer:

The third sequence.  

Step-by-step 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:

• $4 - 12 = -8$.
• $\displaystyle \frac{4}{3} - 4 = -\frac{8}{3}$.
• $\displaystyle \frac{16}{3} - \frac{4}{3} = \frac{12}{3} = 4$.

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

Third option:

• $\displaystyle -\frac{1}{2} - \frac{1}{2} = -1$.
• $\displaystyle -\frac{3}{2} - \left(-\frac{1}{2}\right) = -\frac{3}{2} + \frac{1}{2} = -1$.
• $\displaystyle -\frac{5}{2} - \left(-\frac{3}{2}\right) = -\frac{5}{2} + \frac{3}{2} = -1$.

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

Fourth option:

• $\displaystyle -\frac{1}{2} - \frac{1}{2} = -1$.
• $\displaystyle \frac{1}{2} - \left(-\frac{1}{2}\right) = \frac{1}{2} + \frac{1}{2} = 1$.
• $\displaystyle -\frac{1}{2}\right - \frac{1}{2} = -1$.

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