# What are the next three terms in the series {5/6,2/3,1/2,1/3,.........}?

Nov 10, 2016

The next three numbers are $\frac{1}{6} , 0 , - \frac{1}{6}$

#### Explanation:

Find the next 3 numbers in the arithmetic sequence $\frac{5}{6} , \frac{2}{3} , \frac{1}{2} , \frac{1}{3.} . .$

The next number, or term, in an arithmetic sequence is formed by adding a number to the previous term. This number $d$ is called the common difference.

To find the common difference, subtract the previous term from a term.

For example, if we subtract the 1st term from the 2nd, $\frac{2}{3} - \frac{5}{6} = - \frac{1}{6}$ we get $d = - \frac{1}{6}$

Subtracting the 2nd term from the 3rd gives
$\frac{1}{2} - \frac{2}{3} = - \frac{1}{6}$ and again $d = - \frac{1}{6}$

In fact, subtracting any term from the next term should give the same or "common" difference.

Now that we know $d = - \frac{1}{6}$, we can find the next term by adding it to the previous term.

In this example, we know the 4th term is $\frac{1}{3}$.

So, the 5th term is $\frac{1}{3} - \frac{1}{6} = \frac{1}{6}$

The 6th term is $\frac{1}{6} - \frac{1}{6} = 0$

The 7th term is $0 - \frac{1}{6} = - \frac{1}{6}$