So term 6 equals term 5 plus term 4. We already know term 5 is 21 and term 4 is 13, so:
x6 = 21 + 13 = 34
Many Rules
One of the troubles with finding 'the next number' in a sequence is that mathematics is so powerful we can find more than one Rule that works.
What is the next number in the sequence 1, 2, 4, 7, ?
Here are three solutions (there can be more!):
Solution 1: Add 1, then add 2, 3, 4, ...
So, 1+1=2, 2+2=4, 4+3=7, 7+4=11, etc...
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, ...
(That rule looks a bit complicated, but it works)
Solution 2: After 1 and 2, add the two previous numbers, plus 1:
Rule: xn = xn-1 + xn-2 + 1
Sequence: 1, 2, 4, 7, 12, 20, 33, ...
Solution 3: After 1, 2 and 4, add the three previous numbers
Rule: xn = xn-1 + xn-2 + xn-3
Sequence: 1, 2, 4, 7, 13, 24, 44, ...
So, we have three perfectly reasonable solutions, and they create totally different sequences.
Which is right? They are all right.
And there are other solutions ...
... it may be a list of the winners' numbers ... so the next number could be ... anything! |
Simplest Rule
When in doubt choose the simplest rule that makes sense, but also mention that there are other solutions.
Finding Differences
Sometimes it helps to find the differences between each pair of numbers ... this can often reveal an underlying pattern.
Here is a simple case:
The differences are always 2, so we can guess that '2n' is part of the answer.
Let us try 2n:
The last row shows that we are always wrong by 5, so just add 5 and we are done:
Rule: xn = 2n + 5
OK, we could have worked out '2n+5' by just playing around with the numbers a bit, but we want a systematic way to do it, for when the sequences get more complicated.
Second Differences
In the sequence {1, 2, 4, 7, 11, 16, 22, ...} we need to find the differences ...
... and then find the differences of those (called second differences), like this:
The second differences in this case are 1.
With second differences we multiply by n22
In our case the difference is 1, so let us try just n22:
Rapidweaver 8 1 4 X 42
n: | 1 | 2 | 3 | 4 | 5 |
---|
Terms (xn): | 1 | 2 | 4 | 7 | 11 |
---|
n22: | 0.5 | 2 | 4.5 | 8 | 12.5 |
---|
Wrong by: | 0.5 | 0 | -0.5 | -1 | -1.5 |
---|
We are close, but seem to be drifting by 0.5, so let us try: n22 − n2
Wrong by 1 now, so let us add 1:
n22 − n2 + 1 | 1 | 2 | 4 | 7 | 11 |
---|
Wrong by: | 0 | 0 | 0 | 0 | 0 |
---|
We did it!
The formula n22 − n2 + 1 can be simplified to n(n-1)/2 + 1
So by 'trial-and-error' we discovered a rule that works:
Rule: xn = n(n-1)/2 + 1
Sequence: 1, 2, 4, 7, 11, 16, 22, 29, 37, ...
Other Types of Sequences
Rapidweaver 8 1 4 X 4 Post Caps
Read Sequences and Series to learn about:
And there are also:
And many more!
Rapidweaver 8 1 4 X 4
In truth there are too many types of sequences to mention here, but if there is a special one you would like me to add just let me know.