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• Level: GCSE
• Subject: Maths
• Word count: 1971

In this investigation I am trying to find a rule for the difference between any consecutive numbers in a sequence. I am going to use a series of algebraic expressions to try and come up with a successful rule that works for every consecutive number I try.

Extracts from this document...

Introduction

Math’s Investigation

Consecutive Numbers

Introduction

In this investigation I am trying to find a rule for the difference between any consecutive numbers in a sequence.  I am going to use a series of algebraic expressions to try and come up with a successful rule that works for every consecutive number I try.

2 Consecutive Numbers

1,2 = 3

2,3 = 5

3,4 = 7

4,5 = 9

1,2,3 = 6

2,3,4 = 9

3,4,5 = 12

4,5,6 = 15

4 Consecutive Numbers

1,2,3,4 = 10

2,3,4,5 = 14

3,4,5,6 = 18

4,5,6,7 = 22

As you can see from these results if I carry on the difference goes up by 4 each time.

5 Consecutive Numbers

1,2,3,4,5 = 15

2,3,4,5,6 = 20

3,4,5,6,7 = 25

4,5,6,7,8 = 30

Difference goes up by 5 each time.

6 Consecutive Numbers

1,2,3,4,5,6 = 21

2,3,4,5,6,7 = 27

3,4,5,6,7,8 = 33

4,5,6,7,8,9 = 39

Difference goes up by 6 each time.

I have chosen to display results from 2-6 consecutive numbers.  I have done this because I think it is an adequate amount of data to find and predict patterns in the sequences.  There is also no need to go any further than 6 because I have noticed a pattern, the pattern is: how ever many numbers there are in the sequence it equals the difference between each answer in that sequence every time…

As you can see above, for 5 Consecutive numbers the difference goes up in 5’s every time, 6 consecutive numbers the difference goes up in 6’s every time, and so on…

2 Consecutive Numbers

 Sequence of Numbers Difference 1,2 3 2,3 5 3,4 7 4,5 9

3 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3 6 2,3,4 9 3,4,5 12 4,5,6 15

4 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3,4 10 2,3,4,5 14 3,4,5,6 18 4,5,6,7 22

5 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3,4,5 15 2,3,4,5,6 20 3,4,5,6,7 25 4,5,6,7,8 30

6 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3,4,5,6 21 2,3,4,5,6,7 27 3,4,5,6,7,8 33 4,5,6,7,8,9 39

Middle

This is how it works: 2 x 1 + 1 = 3

2 x 2 + 1 = 5

2 x 3 + 1 = 7

2 x 4 + 1 = 9

3 Consecutive Numbers

1              2                3                4

6        9        12        15

3        3        3

I found that the rule for this sequence of numbers is 3n+3

This is how it works: 3 x 1 + 3 = 6

3 x 2 + 3 = 9

3 x 3 + 3 = 12

3 x 4 + 3 = 15

4 Consecutive Numbers

1        2        3        4

10        14        18        22

4        4        4

In this sequence the difference is 4 every time that means it has something to do with 4n.  I followed the same steps as I did in the previous two sequences to try and find a rule:

I found that the rule for 4 consecutive numbers = 4n+6

How it works: 4 x 1 + 6 = 10

4 x 2 + 6 = 14

4 x 3 + 6 = 18

4 x 4 + 6 = 22

5 Consecutive Numbers

1        2        3        4

15        20        25        30

5        5        5

Rule for 5 Consecutive Numbers = 5n+10

How it works: 5 x 1 + 10 = 15

5 x 2 + 10 = 20

5 x 3 + 10 = 25

5 x 4 + 10 = 30

6 Consecutive Numbers

1        2        3        4

21        27        33        39

6        6        6

Rule for 6 Consecutive Numbers = 6n+15

How it works: 6 x 1 + 15 = 21

6 x 2 + 15 = 27

6 x 3 + 15 = 33

6 x 4 + 15 = 39

5 Consecutive Numbers = n+n+1+n+2+n+3+n+4 = 5n+10

Conclusion

xn + ½ x² - ½ x

Extension

To extend this investigation further I am going to investigate multiplying consecutive numbers together.  For example when you times three consecutive whole numbers together is the answer always divisible by 3? I am going to do this for 4, 5 and 6 consecutive numbers to get a various array of data.

3 Consecutive Numbers

1,2,3 = 6

2,3,4 = 24

3,4,5 = 60

4,5,6 = 120

6        24        60        120

All these numbers are divisible by 3.

4 Consecutive Numbers

1,2,3,4 = 24

2,3,4,5 = 120

3,4,5,6 = 360

4,5,6,7 = 840

24        120        360        840

All these numbers are divisible by 4.

5 Consecutive Numbers

1,2,3,4,5 = 120

2,3,4,5,6 = 720

3,4,5,6,7 = 2520

4,5,6,7,8 = 6720

120        720        2520        6720

For 5 consecutive numbers all the answers are divisible by 5.

6 consecutive Numbers

1,2,3,4,5,6 = 720

2,3,4,5,6,7 = 5040

3,4,5,6,7,8 = 20160

4,5,6,7,8,9 = 60480

720        5040        20160        60480

All these numbers are divisible by 6.

From these results I have collected I have noticed when you multiply any given amount of consecutive numbers together the answer to that sum will always be divisible by the amount of consecutive numbers in that sequence.

Table of Results

3 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3 6 2,3,4 24 3,4,5 60 4,5,6 120

4 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3,4 24 2,3,4,5 120 3,4,5,6 360 4,5,6,7 840

5 Consecutive Numbers

 Sequence of Numbers Difference 1,2,3,4,5 120 2,3,4,5,6 720 3,4,5,6,7 2520 4,5,6,7,8 6720

6 consecutive Numbers

 Sequence of Numbers Difference 1,2,3,4,5,6 720 2,3,4,5,6,7 5040 3,4,5,6,7,8 20160 4,5,6,7,8,9 60480

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