# How do you write a rule for the nth term of the geometric sequence given the two terms a_3=5, a_6=5000?

Mar 19, 2017

Rule for the ${n}^{t h}$ term of the geometric sequence is ${a}_{n} = {10}^{n - 1} / 20$

#### Explanation:

If ${m}^{t h}$ term of a geometric sequence is ${a}_{m}$ and ${n}^{t h}$ term of the sequence is ${a}_{n}$, then the common ratio $r$ is given by

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

Here, we have ${a}_{3} = 5$ and ${a}_{6} = 5000$, hence

$r = {\left(\frac{5000}{5}\right)}^{\frac{1}{\left(6 - 3\right)}} = {1000}^{\frac{1}{3}} = \sqrt[3]{1000} = 10$

and as ${n}^{t h}$ term of a geometric sequence is given by

${a}_{n} = {a}_{1} \times {r}^{n - 1}$ and hence as ${a}_{3} = 5$

${a}_{1} = {a}_{n} / {r}^{n - 1} = \frac{5}{10} ^ 2 = 0.05$

and rule for the ${n}^{t h}$ term of the geometric sequence is

${a}_{n} = 0.05 \times {10}^{n - 1} = {10}^{n - 1} / 20$