# What are the first three terms of a geometric sequence with a common ratio of 2/5 if the sixth term is 64/25?

Jul 26, 2016

${a}_{1} = 250 , {a}_{2} = 100 , {a}_{3} = 40$

#### Explanation:

The ${n}^{t h}$ term of a geometric sequence is given by:

${a}_{n} = {a}_{1} {r}^{n - 1}$ Where $r$ is the Common Ratio and ${a}_{1}$ is the ${1}^{s t}$ term.

In this example; $r = \frac{2}{5} , n = 6$
Thus, ${a}_{6} = {a}_{1} {r}^{6 - 1}$

We are told that ${a}_{6} = \frac{64}{25} \to \frac{64}{25} = {a}_{1} \cdot {\left(\frac{2}{5}\right)}^{5}$

Hence, ${a}_{1} = \frac{64}{25} \cdot {5}^{5} / {2}^{5} = {2}^{6} / {5}^{2} \cdot {5}^{5} / {2}^{5}$

${a}_{1} = {2}^{1} \cdot {5}^{3} = 2 \cdot 125 = 250$

${a}_{2} = 250 \cdot \frac{2}{5} = 100$
${a}_{3} = 100 \cdot \frac{2}{5} = 40$