# A geometric sequence is defined recursively by a_n= 4a_(n-1), and the first term of the sequence is 0.5, how do you find the explicit formula?

Jul 3, 2018

${a}_{n} = 0.5 {\left(4\right)}^{n - 1}$

#### Explanation:

Looking at this we have...
${a}_{n} = 4 {a}_{n - 1}$

${a}_{2} = 4 {a}_{2 - 1}$
${a}_{2} = 4 {a}_{1}$
${a}_{2} = 4 \cdot 0.5$
${a}_{2} = 2$

Since we are told that is a geometric sequence there has to be a $r$ or a common ratio:
$r = {a}_{2} / {a}_{1}$
$r = \frac{2}{.5} = 4$

So explicitly written, the formula would be:
${a}_{n} = 0.5 {\left(4\right)}^{n - 1}$