Arithmetic Sequences #1 (Algebra)

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 $a_n=2n\:-\:1$.

Since the first term’s numerator and denominator are the same, we can easily combine the formula as follows:
$$a_n=\frac{n}{2n\:-\:1}$$

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