### Arithmetic Sequences #1 Question:

##### Find the explicit formula for the following sequence: 1, 2/3, 3/5, 4/7.

#### Answer:

To get the explicit formula for the arithmetic sequences #1 question, we could express the first term 1 as $\frac{1}{1}$.

By looking at the sequence as $$\frac{1}{1},\:\frac{2}{3},\:\frac{3}{5},\:\frac{4}{7}$$, the patterns could be distinguished from the numerator and from the denominator separately.

The terms at the **numerator** have the following characteristics:

- First term = 1
- Common difference = 2 – 1 = 3 – 2 = 1

Therefore, the arithmetic sequence formula could be used to get the general formula for the numerator.

$$a_n=a_1+\left(n-1\right)d$$

$$a_n=1+\left(n-1\right)1$$

This could be simplified as $a_n=n$ for the numerator.

Likewise, the denominator has the same first term as the numerator. However, the numerator and the denominator differ in the common difference. The terms at the **denominator** have the following properties:

- First term = 1
- Common difference = 3 – 1 = 5 – 3 = 2

By plugging this values at the arithmetic sequence formula, we would have the following:

$$a_n=a_1+\left(n-1\right)d$$

$$a_n=1+\left(n-1\right)2$$

This could be simplified as $a_n=2n\:-\:1$.