# How do you write a rule for the nth term of the geometric sequence given the two terms a_2=36, a_4=576?

May 16, 2017

See explanation.

#### Explanation:

For any geometric sequence ${a}_{n}$, and natural $n , k$ $\left(n > k\right)$ you can write that:

## ${a}_{n} = {a}_{k} \cdot {q}^{n - k}$

In this task we can write that:

## $576 = 36 {q}^{2}$

${q}^{2} = 16$

This leads to 2 possible values of $q$: ${q}_{1} = - 4$ and ${q}_{2} = 4$

Now we can calculate the first term separately for $q = - 4$ and $q = 4$

If $q = - 4$ then ${a}_{1} = \frac{36}{- 4} = - 9$

else if $q = 4$ then ${a}_{1} = \frac{36}{4} = 9$

So this task has 2 solutions:

${a}_{n} = \left(- 9\right) \cdot {\left(- 4\right)}^{n - 1}$ or ${a}_{n} = 9 \cdot {4}^{n - 1}$