# Given that the third term a_3 of a geometric sequence is 2 and the sixth term a_6 is 128, which term is 1/2 ?

Jul 26, 2016

${a}_{2}$

#### Explanation:

We can write the general term of a geometric sequence as:

${a}_{n} = a {r}^{n - 1}$

where $a$ is the initial term and $r$ is the common ratio.

In our example ${a}_{3} = a {r}^{2}$ and ${a}_{6} = a {r}^{5}$ so we find:

${r}^{3} = \frac{a {r}^{5}}{a {r}^{2}} = {a}_{6} / {a}_{3} = \frac{128}{2} = 64 = {4}^{3}$

So the only possible Real value of $r$ is $\sqrt[3]{{4}^{3}} = 4$

Note that $\frac{1}{2} = \frac{2}{4} = {a}_{3} / r$.

So ${a}_{2} = \frac{1}{2}$