Series: 1 2 4 7 12 20 .. Differences 1 2 3 5 8 ..and, if you've been following these pages closely, you should recognize the differences! However, this doesn't help us discover a pattern for the series itself.
If the differences are constant, that is, all the same,
then the series is related to the square numbers
Series: 1 2 4 7 12 20 .. 1 + 2 = 3 2 + 4 = 6 4 + 7 =11 7 + 12=19It looks like we have the sum of the previous two terms gives one less than the next term, or, in maths notation:
Fibonacci: 0 1 1 2 3 5 8 13 21 .. Differences: 1 0 1 1 2 3 5 8 ..we notice these are also the Fibonacci numbers. A moments thought should convince you that this must be true since we get the next Fibonacci number from the previous one by... adding in the Fibonacci number before that one.
Now we can see how our original series (1, 2, 4, 7, 12, 20,..) is made. Since its differences are the same as the Fibonacci series differences, we can add or subtract a constant to the Fibonacci series. If we add 1 to the each term of the series we get 2, 3, 5, 8, 13, 21.
The problem remaining is "Is it Fib(n)1 or Fib(n+1)1 or Fib(n1)1, or what?".
n: 1 2 3 4 5 6 .. Series: 1 2 4 7 12 20 .. Fib(n): 1 1 2 3 5 8 .. so series is s(n) = Fib(n+2)1since each term of the series is one less than the Fibonacci number two further along.
