# How do you determine whether each sequence could be geometric, arithmetic, or neither. Use a patte to write the next three terms. 460, 46, 4.6, 0.46?

Feb 16, 2018

See below.

#### Explanation:

If a, b, c are in arithmetic sequence then:

$b - a = c - b$

This is known as the common difference

If a, b, c are in geometric sequence then:

$\frac{b}{a} = \frac{c}{b}$

This is known as the common ratio

From given sequence:

$460 , 46 , 4.6 , 0.46$

$b - a = c - b$

$46 - 460 = - 414$

$4.6 - 46 = - 41.4$

$- 414 \ne - 41.4 \textcolor{w h i t e}{88}$ not an arithmetic sequence

$\frac{b}{a} = \frac{c}{b}$

$\frac{46}{460} = 0.1$

$\frac{4.6}{46} = 0.1 \textcolor{w h i t e}{88}$This is a geometric sequence

The nth term of a geometric sequence is given by:

$a {r}^{n - 1}$

Where $\boldsymbol{a}$ is the first term, $\boldsymbol{r}$ is the common ratio and $\boldsymbol{n}$ is the $\boldsymbol{n t h}$ term.

From example:

$\boldsymbol{a} = 460$

$\boldsymbol{r} = 0.1$

We need to find the next 3 terms. i.e. 5th 6th and 7th.

So:

$5 \text{th} = 460 {\left(0.1\right)}^{4} = 0.046$

$6 \text{th} = 460 {\left(0.1\right)}^{5} = 0.0046$

$7 \text{th} = 460 {\left(0.1\right)}^{6} = 0.00046$

We can see from the pattern, that as the sequence progresses each number is 10 times smaller than the previous number.