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Extrapolation
If we encounter the sequence 1, 2, 4, 8, 16, ... we expect that the next number will be 32, because these are apparently powers of 2.
But how sure is that?
Explanation
The formula is apparently
and the corresponding table then shows
n = 1 1 2 2 3 4 4 8 5 16 6 32
There are other possibilities to extend the original sequence. For example, the number 31 might also be correct. Then we do need another formula. For that we use here
and substitute for x the values 1, 2, 3, 4, 5 and 6. The table becomes then
x = 1 1 2 2 3 4 4 8 5 16 6 31
That's funny. That looks like hocus pocus. You can't just come up with a formula like that. What's behind it, and where does that formula come from?
Calculation
From the fundamental theorem of mathematics it follows that you can describe a curve that goes through 5 points with a fourth degree function. Therefore we start with
and take for x the values 1, 2, 3, 4 and 5. Then you get
a
b
c
d
e
=
1 1 1 1 1 1 16 8 4 2 1 2 81 27 9 3 1 4 256 64 16 4 1 8 625 125 25 5 1 16
Here we eliminate e, and get
a
b
c
d
=
15 7 3 1 1 80 26 8 2 3 255 63 15 3 7 624 124 24 4 15
Now we eliminate d, and get
a
b
c
=
50 12 2 1 210 42 6 4 564 96 12 11
Then we eliminate c, and get
a
b
=
60 6 1 264 24 5
Finally, we eliminate b, and get
a
=
24 1
That means
and substitute that
Next we find
Then comes
And finally
The formula then becomes
and we write that here as
We check that, and calculate the table
However, the sixth number must be 32. So it is 1 too little. The deviation increases, because
and that should have been 64. That is already a difference of 7.
Example 1
On a Casio fx-9860GII calculator, you can see that the simultaneous linear equations with 5 unkowns
a b c d e f 1 1 1 1 1 1 16 8 4 2 1 2 81 27 9 3 1 4 256 64 16 4 1 8 625 125 25 5 1 16
give
1 24