# How do you write the expression for the nth term of the geometric sequence a_3=16/3, a_5=64/27, n=7?

May 14, 2018

${a}_{n} = {2}^{n + 1} / {3}^{n - 2}$
So:
${a}_{7} = \frac{256}{243}$

#### Explanation:

${a}_{3} = \frac{16}{3} , {a}_{5} = \frac{64}{27}$

${a}_{5} = \frac{16 \cdot 2 \cdot 2}{3 \cdot 3 \cdot 3} = {a}_{3} \cdot \frac{2}{3} \cdot \frac{2}{3}$

So: ${a}_{n + 1} = {a}_{n} \cdot \frac{2}{3}$

That will mean:
${a}_{3} = {a}_{2} \cdot \frac{2}{3} \Rightarrow {a}_{2} = {a}_{3} / \left(\frac{2}{3}\right) = \frac{\frac{16}{2}}{\frac{3}{3}} \Rightarrow {a}_{2} = \frac{8}{1}$

So, we have:
${a}_{2} = {2}^{3} / {3}^{0} , {a}_{3} = {2}^{4} / {3}^{1} , {a}_{5} = {2}^{6} / {3}^{3}$

We can deduct:
${a}_{n} = {2}^{n + 1} / {3}^{n - 2}$

So:
${a}_{7} = {2}^{8} / {3}^{5} = \frac{256}{243}$