# How do you find the explicit formula for the following sequence 1/2,3/4,5/8,7/16 ...?

May 30, 2016

${a}_{n} = \frac{2 n - 1}{2} ^ n$

#### Explanation:

The numerators form an arithmetic sequence:

$1 , 3 , 5 , 7$

with common difference $2$.

So a formula for the numerator could be written:

${p}_{n} = 1 + 2 \left(n - 1\right) = 2 n - 1$

The denominators form a geometric sequence:

$2 , 4 , 8 , 16$

with common ratio $2$.

So a formula for the denominator could be written:

${q}_{n} = 2 \cdot {2}^{n - 1} = {2}^{n}$

Thus a formula for a general term of our example sequence can be written:

${a}_{n} = {p}_{n} / {q}_{n} = \frac{2 n - 1}{2} ^ n$