# How do you write an nth term rule for a_3=10 and a_6=300?

Feb 23, 2018

${a}_{n} = 1.0359 \cdot {3.107}^{n - 1}$

#### Explanation:

I'm assuming this is a geometric sequence...

The general formula for an arithmetic sequence is

${a}_{n} = {a}_{1} \cdot {r}^{n - 1}$ with ${a}_{1}$ as the first term and $r$ as the common ratio.

Plug in:

${a}_{3} = 10 = {a}_{1} \cdot {r}^{3 - 1}$
${a}_{6} = 300 = {a}_{1} \cdot {r}^{6 - 1}$

$10 = {a}_{1} \cdot {r}^{2}$
$300 = {a}_{1} \cdot {r}^{5}$

Divide:

$\frac{300}{10} = \frac{{a}_{1} \cdot {r}^{5}}{{a}_{1} \cdot {r}^{2}}$

$30 = \frac{\cancel{{a}_{1}} \cdot {r}^{5}}{\cancel{{a}_{1}} \cdot {r}^{2}}$

$30 = {r}^{3}$

$\sqrt[3]{30} = r \mathmr{and} {30}^{\frac{1}{3}}$

$r \approx 3.107$

Then find ${a}_{1}$:

$10 = {a}_{1} \cdot 9.653$

${a}_{1} \approx 1.0359$

Plug in the information into the general formula:

${a}_{n} = 1.0359 \cdot {3.107}^{n - 1}$

Note: This is not exact as I rounded somewhere.