# How do you write a rule for the nth term of the geometric term given the two terms a_2=2, a_5=1/4?

May 22, 2017

${a}_{n} = 4 \cdot {\left(\frac{1}{2}\right)}^{n - 1}$

#### Explanation:

The general form for a geometric sequence is:
${a}_{n} = {a}_{1} \cdot {r}^{n - 1}$

With this, you can write a system of equations:
${a}_{2} = {a}_{1} \cdot {r}^{2 - 1}$
${a}_{5} = {a}_{1} \cdot {r}^{5 - 1}$

$2 = {a}_{1} \cdot {r}^{1}$
$0.25 = {a}_{1} \cdot {r}^{4}$
${r}^{4} / {r}^{1} = \frac{0.25}{2}$
${r}^{3} = 0.125$
$r = 0.5$

Now, find the first term.
${a}_{2} = {a}_{1} \cdot {0.5}^{1}$
$2 = {a}_{1} \cdot 0.5$
${a}_{1} = 4$

Using the formula stated above, we get:
${a}_{n} = 4 \cdot {0.5}^{n - 1}$