# Working with Sequences on a Graphing Calculator

## Key Questions

You can$t really do that. #### Explanation: I$m not sure what you want is possible unless you were given the formula for that sequence. Otherwise you'll have to figure out the relevant formula before you plug it into the calculator.

Example: If I tell you to give me the 10th number in the sequence:
$2 , 5 , 10 , 17 , 26 , \ldots$ YOU have to figure out that the "formula" for that sequence is $f \left(n\right) = {n}^{2} + 1$ and then you can plug ${10}^{2} + 1$ in the calculator and find the answer. So unless you get the formula, the hard work (finding it) is up to you.

Assuming the terms are nonzero, we can find the common ratio $r$ on a calculator by taking any two consecutive terms and dividing the later one by the earlier one:

$r = {a}_{n + 1} / {a}_{n}$

#### Explanation:

A geometric sequence is a sequence with a common ratio $r$ between adjacent terms, that is, a sequence of the form ${a}_{1} , {a}_{1} r , {a}_{1} {r}^{2} , \ldots , {a}_{1} {r}^{n} , \ldots$

Then, assuming the terms are nonzero, dividing any term by the prior term will give the common ratio:

$\frac{\cancel{{a}_{1}} {r}^{n}}{\cancel{{a}_{1}} {r}^{n - 1}} = {r}^{n} / {r}^{n - 1} = {r}^{n - \left(n - 1\right)} = {r}^{1} = r$

To find $r$ on a calculator, then, take any two consecutive terms and divide the later one by the earlier one.

In fact, more generally, given any two terms ${a}_{1} {r}^{m}$ and ${a}_{1} {r}^{n}$, $m < n$, we can find $r$ by dividing $\frac{{a}_{1} {r}^{n}}{{a}_{1} {r}^{m}}$ and taking the ${\left(n - m\right)}^{\text{th}}$ root:

${\left(\frac{{a}_{1} {r}^{n}}{{a}_{1} {r}^{m}}\right)}^{\frac{1}{n - m}} = {\left({r}^{n - m}\right)}^{\frac{1}{n - m}} = {r}^{\frac{n - m}{n - m}} = {r}^{1} = r$