How do you write a rule for the nth term of the geometric sequence given the two terms a_2=4, a_5=256/27?

Jun 26, 2018

${a}_{n} = 3 \cdot {\left(\frac{4}{3}\right)}^{n - 1}$

Explanation:

The geometric ${n}^{\text{th}}$ term is ${a}_{n} = a {r}^{n - 1}$

Where $a$ is the first term

$a r = 4$

$a {r}^{4} = \frac{256}{27}$

$\implies a = \frac{4}{r}$

$\implies \frac{4}{r} \cdot {r}^{4} = \frac{256}{27}$

$\implies 4 {r}^{3} = \frac{256}{27}$

$\implies r = \frac{4}{3}$

$\implies a = \frac{4}{r} = \frac{4}{\frac{4}{3}} = 3$

$\implies {a}_{n} = 3 \cdot {\left(\frac{4}{3}\right)}^{n - 1}$