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What is the sum of nth term of a sequence:
3, 4, 7, 11, 18, 29 .... nth (fibonacci)

Due to the difficulty of writing equations here, I've included links to illustrations.

The terms of your sequence are 3, 4, 7, 11, 18, 28, ...
If you compare this with the Fibonacci sequence, you'll see that
the nth term of this sequence is the sum of the nth and (n+2)th Fibonacci numbers.

Some facts about the Fibonacci sequence:
The nth term is (φⁿ - (1-φ)ⁿ/√5, where φ is the "golden ratio" (1+√5)/2.
The sum of the first n terms equals the (n+2)th term minus 1.

The sum of the first n terms of your sequence can be expressed in terms of the Fibonacci numbers:
  ∑t = [2(n+4)th number] - [(n+1)th number] - [(n+2)th number] - 4
which you can then calculate using φ. You can see the calculations here:
The final equation looks a fright!--but I have verified its accuracy in an Excel spreadsheet.  

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Janet Yang


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