# Fraction Differences

Fraction Differences

First Sequence

To begin with I looked at the first sequence of fractions to discover the formula that explained it. As all the numerators were 1 I looked at the denominators. As these all increased by 1 every time, I figured that the formula was simply as the denominators corresponded to the implied first line as shown in this table below:

nth number

2

3

4

5

6

7

8

Denominators

2

3

4

5

6

7

8

I shall call this Formula 1 (F1) for easy reference.

Second Sequence

Again I decided to discount the numerator as it was 1, and I decided to concentrate on the differences between the denominators rather than the 'fractions'. So I am looking for a formula that will explain the sequence: 2, 6, 12, 20, 30.

First of all though I decided to extend the sequence in order to have a broader range to work with. I used a calculator to work out the following denominators finding the difference between and , and all the way up to

I set the differences out in a table to try to find the pattern:

nth number

2

3

4

5

6

7

8

9

Sequence

2

6

2

20

30

42

56

72

90

First Difference

4

6

8

0

2

4

6

8

Second Difference

2

2

2

2

2

2

2

2

As there was a constant difference of 2 I believed that the formula would include n². I applied this to the first number in the sequence '2'. So n² = (1 x 1 = 1). To get the first number of the sequence - 2 I would have to add 1. Therefore the formula could be: n2 + 1

I tried this formula for the second number in the series. 22 = 2 x 2 = 4 + 1 = 5.

My formula was wrong.

As I knew that the formula would include n² I looked at the difference between n² and the numbers in the sequence:

Sequence

Difference

2

x 1

+ 1

6

2 x 2

+ 2

2

3 x 3

+ 3

20

4 x 4

+ 4

30

5 x 5

+ 5

By looking at this I saw that the formula was n² + n.

The formula n² + n ...