How to Quickly Add 10 Numbers in the Fibonacci Number Pattern


The Fibonacci sequence is one of the most famous in mathematics. You can quickly add ten consecutive numbers in the Fibonacci sequence using a simple trick, which we will also prove.

In short: take the 7th of the 10 consecutive numbers in the Fibonacci sequence and multiply that by 11.

Part 1
The Trick

The Trick on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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Here are two examples to follow along: Starting at the beginning of the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. The 7th number here is 8. Starting a little further down the sequence: 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. The 7th number here is 89.

The Trick on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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0, 1, 1, 2, 3, 5, 8, 13, 21, 34: the 7th number was 8, multiplying that by 11 gives us 8*11=88 5, 8, 13, 21, 34, 55, 89, 144, 233, 377: the 7th number was 89, multiplying that by 11 gives us 89*11=979

The Trick on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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You can also add the 10 numbers manually to check your work. 1+1+2+3+5+8+13+21+34=88 5+8+13+21+34+55+ 89+144+233+377=979

Part 2
Proof

Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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The nth term is defined to be the sum of the (n-1)th and (n-2)th term, with the first two terms being 0 and 1.

Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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Continue this pattern to define the rest of our 10 numbers that we want to add: f(n+3) = a+2b f(n+4) = 2a+3b f(n+5) = 3a+5b f(n+6) = 5a+8b f(n+7) = 8a+13b f(n+8) = 13a+21b f(n+9) = 21a+34b

Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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The sum of f(n)+...+f(n+9) = 55a+88b.

Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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Thus, 11*(5a+8b)=11*f(n+6)

Proof on How to Quickly Add 10 Numbers in the Fibonacci Number Pattern

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f(n)+...+f(n+9)=11*f(n+6).