# How do you write a rule for the nth term of the geometric sequence and then find a_5 given a_4=189/1000, r=3/5?

Feb 23, 2017

${a}_{5} = \setminus \frac{7}{8} \setminus \times {\left(\setminus \frac{3}{5}\right)}^{4} = \setminus \frac{567}{500}$

#### Explanation:

We know we can write every geometric sequence in the form of below:

${a}_{n} = {a}_{1} \setminus \times {r}^{n - 1}$

and for now what we have to do is to figure out what is our first term ( ${a}_{1}$) and we can do it easily cause we have ${a}_{4}$:

${a}_{4} = \setminus \frac{189}{1000} = {a}_{1} \setminus \times {\left(\setminus \frac{3}{5}\right)}^{3}$

and we have to solve this equation for ${a}_{1}$

${a}_{1} = \setminus \frac{\setminus \frac{189}{1000}}{\setminus \frac{{3}^{3}}{{5}^{3}}} = \setminus \frac{189 \setminus \times 125}{1000 \setminus \times 27} = \setminus \frac{7}{8}$

Now we can rewrite our equation for any nth term:

${a}_{n} = {a}_{1} \setminus \times {r}^{n - 1} = \setminus \frac{7}{8} \setminus \times {\left(\setminus \frac{3}{5}\right)}^{n - 1}$

and we can calculate ${a}_{5}$ just by putting our $n = 5$ in the equation.

${a}_{5} = \setminus \frac{7}{8} \setminus \times {\left(\setminus \frac{3}{5}\right)}^{4} = \setminus \frac{567}{500}$