For example:
5 Consecutive Numbers
1,2,3,4,5 = 15
3 x 5 = 15
This rule works for both odd and even numbers, as you can see below:
6 Consecutive Numbers
1,2,3,4,5,6 = 21
3.5 x 6 = 21
Algebraic Rules
From the results I have collected I have found an algebraic rule:
While I was studying my results I noticed a pattern in one of the sequences, I tried this rule I had found on all the other consecutive numbers and it worked…
2 Consecutive Numbers
3 5 7 9
These numbers are the first four answers in the 2 consecutive numbers table.
While I was looking at these numbers I found a pattern. The difference between each of the numbers is 2 every time, this tells me that it has got something to do with 2n.
Once I knew 2n was involved I could try to figure out what number I could add on to 2n to equal the answer.
1 2 3 4
3 5 7 9
2 2 2
I found that the rule for this sequence of numbers is 2n+1
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
You can also find the rule for any consecutive numbers by adding the numbers below together…
2 Consecutive Numbers = n+n+1 = 2n+1
3 Consecutive Numbers = n+n+1+n+2 = 3n+3
4 Consecutive Numbers = n+n+1+n+2+n+3 = 4n+6
5 Consecutive Numbers = n+n+1+n+2+n+3+n+4 = 5n+10
6 Consecutive Numbers = n+n+1+n+2+n+3+n+4+n+5 = 6n+15
Table Of Results
From the results table above I have noticed a pattern in the number that you add to n to equal the difference for any consecutive number.
2 3 4 5 6
1 3 6 10 15
3 4 5 6
1 1 1
Because the 2nd line of differences are the same this tells me that it has got something to do with n². Because I know it has got something to do with n² I can now use this formula to try and work out a rule…
ax² + bx + c
x = 2 4a + 2b + c = 1
- 5a + b = 2
x = 3 9a + 3b + c = 3 - 2a = 1 a = ½
- 7a + b = 3
x = 4 16a + 4b + c = 6
To work out “b” I have to put a’s answer into this sum and work it out…
5 x ½ + b = 2
2.5 + b = 2
b = -½
Now I have worked out “a” and “b’s” value’s I can now work out the value of c… I do this by trying to find the number needed to make this sum correct.
4a + 2b + c = 1
4 x ½ + 2 x -½ + c = 1
2 – 1 + 0 = 1
c = 0
From these results I have just collected I can now work out a rule. The rule is:
To prove this rule works I am going to check it…
I am going to check it using 3 and 5consecutive numbers, I’m doing this because I know the answer for both 3 and 5 consecutive numbers, so I can check if I got it correct…
3 consecutive numbers
3n + 3² 2 - 1½
3n + 4½ - 1½ = 3n+3
3n+3
5 Consecutive Numbers
5n + 5² 2 – 2½
5n + 12½ - 2½ = 5n+10
5n+10
By using both 3 and 5 consecutive numbers to check my rule I have proved that it works…
Final Table of Results
Now I’ve got a general rule I am going to do some predictions to try and back it up.
Predictions
10 Consecutive Numbers
10n + 10² 2 – 5
10n + 50 – 5
10n + 45
For 10 consecutive numbers I predict the rule to be 10n+45. To make sure this rule is correct I am going to use the long way to try and find it…
10 Consecutive Numbers = n+n+1+n+2+n+3+n+4+n+5+n+6+n+7+n+8+n+9 = 10n + 45
And as you can see from the results above my prediction was correct.
By using this simple rule I can predict the difference of any amount of consecutive numbers in any given sequence.
Conclusion
Throughout this investigation I have been trying to find a general rule that works for any sequence of consecutive numbers given. During the beginning of this investigation I thought I found a rule that would work for any number of consecutive numbers given, but I was wrong. The rule that I had was fine if I knew what the numbers were, but if I didn’t it wasn’t very useful. I needed a rule that would work for x amount of consecutive numbers. After a lot of testing and evaluating I finally came up with a rule that would work…
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
4 Consecutive Numbers
5 Consecutive Numbers
6 consecutive Numbers
