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How can I find the sum of fibonacci (1) + (2) + (4) + (7) + (11) + (16) + ... ?

asked 2018-08-18 01:21:34 -0500

pizza gravatar image

How can I find the sum of fibonacci (1) + (2) + (4) + (7) + (11) + (16) + ... ?

  • The numbers in () means the terms of fibonacci sequence
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answered 2018-08-18 08:46:09 -0500

slelievre gravatar image

updated 2018-09-02 13:44:06 -0500

Sage provides fibonacci to access elements of the Fibonacci sequence.

This can be combined with basic list comprehension and sum.

So, you could do things like listing or summing consecutive terms:

sage: [fibonacci(j) for j in (0 .. 12)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144]
sage: sum(fibonacci(j) for j in (0 .. 12))
376

or non-consecutive terms:

sage: [fibonacci(j) for j in (1, 2, 4, 7, 11, 16)]
[1, 1, 3, 13, 89, 987]
sage: sum(fibonacci(j) for j in (1, 2, 4, 7, 11, 16))
1094

It seems the indices you are interested about are of the form $(n^2 + n + 2)/2$

sage: [(n^2 + n + 2)//2 for n in range(10)]
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

so here are the terms of the Fibonacci sequence for these indices:

sage: [fibonacci((n^2 + n + 2)//2) for n in range(10)]
[1, 1, 3, 13, 89, 987, 17711, 514229, 24157817, 1836311903]

and here is their sum them up to some point:

sage: sum(fibonacci((n^2 + n + 2)//2) for n in range(10))
1861002754

and here is how this sum evolves when you push it further and further:

sage: [sum(fibonacci((j^2 + j + 2)//2) for j in range(n)) for n in range(10)]
[0, 1, 2, 5, 18, 107, 1094, 18805, 533034, 24690851]

Since all the Fibonacci numbers are at least one, if you add infinitely many of them, the sum will tend to infinity.

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Comments

I appreciate you for helping me.

However, what I want to do is not adding up continuous terms of the sequence. I want to add the 1st, 2nd, 4th, 7th, 10th, ... positive fibonacci numbers, which you would notice, +1 +2 +3 +4 +5

I know that this is not quite an easy question, so if you have time you could just give it a try. THANK YOU!

pizza gravatar imagepizza ( 2018-08-20 01:19:36 -0500 )edit

I expanded my answer to address your comment.

slelievre gravatar imageslelievre ( 2018-09-02 13:45:46 -0500 )edit

Thank you!

pizza gravatar imagepizza ( 2018-09-03 03:10:16 -0500 )edit
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answered 2018-08-19 07:27:53 -0500

tmonteil gravatar image

updated 2018-08-19 08:21:42 -0500

The sum is infinite ! You have to ask more precise questions if you want an efficient answer. For exremely large numbers, you should notice that the fibonacci sequence is the difference between two geometric functions.

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Yes, I suppose that the question also wants me to add the numbers to infinity. But I guess adding up a certain amount of terms might be enough to estimate the sum of infinity of this sequence.

pizza gravatar imagepizza ( 2018-08-20 01:24:44 -0500 )edit

Moreover, as I am a beginner in using sage programme. and I am not advance in maths (Sorry!), could you explain what kind of precise questions I should ask for you to answer my question? Thanks!

pizza gravatar imagepizza ( 2018-08-20 01:29:00 -0500 )edit

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Asked: 2018-08-18 01:21:34 -0500

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Last updated: Sep 02