# How do you find the next two terms in the sequence 40, 10 20 50?

Aug 22, 2017

They could be $80$ and $110$, with the general term of the sequence given by:

${a}_{n} = \left\mid 70 - 30 n \right\mid$

but without more given information, they could actually be anything.

#### Explanation:

Given the sequence:

$40 , 10 , 20 , 50$

Note that no finite initial sequence determines the following terms, unless you are given more information about the nature of the sequence (e.g. arithmetic, geometric, harmonic,...).

In the given example, there does not appear to be a simple pattern, apart from the fact that all of the terms are divisible by $10$.

It may be that minus signs have been omitted from the third and fourth terms, since:

$40 , 10 , - 20 , - 50$

is an arithmetic progression.

Perhaps the minus signs have been deliberately removed by taking absolute values.

The $n$th term ${t}_{n}$ of an arithmetic progression can be described by the formula:

${t}_{n} = a + d \left(n - 1\right)$

where $a$ is the initial term and $d$ the common difference.

For example, with $a = 40$ and $d = - 30$ we get the formula:

${t}_{n} = 40 - 30 \left(n - 1\right) = 70 - 30 n$

describing the sequence:

$40 , 10 , - 20 , - 50$

So to describe the given example, we just need to take absolute values:

${a}_{n} = \left\mid 70 - 30 n \right\mid$

If this is the intended sequence, then the next two terms are: $80$ and $110$.