Leapfrog Addition of Fibonacci sequence and the Golden (Ratio) prediction

This trick combines a quick mental calculation with an astonishing prediction.

The Fibonacci sequence is a series of whole numbers in which each number is the sum of the two preceding numbers. Beginning with 0 and 1, the sequence would be 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… etc.

 
You can easily amaze your friends and fellows by telling them the sum of any 10 Fibonacci sequence numbers at a glance.

First, tell the spectator to take a Fibonacci sequence of 10 numbers (as above) behind you. This needs them to write down the numbers sequentially. Now, have the spectator show you the sequence for once and you can easily tell them the sum of those 10 numbers. How? To perform the quick calculation, you notice the 7th number of the sequence and simply multiply the number by11 mentally. So, it's just a matter of blink of eyes!

For example, say your spectator chooses 5 for the first number and 6 for the second number. Then have him get a Fibonacci sequence. 5+6 gives the 3rd number which is 11; 6+11 gives the 4th number which is 17. The entire sequence is as follows:

1^st      - 5

2^nd     - 6

3^rd      - 11

4^th      - 17

5^th      - 28

6^th      - 45

7^th      - 73

8^th      - 118

9^th      - 191

10^th    - 309

In the above example, you could instantly announce that the numbers sum up to 803 faster than the spectator could do using a calculator! (The 7th number is 73. Multiply 73 by 11 and the answer is 803. If you add all 10 numbers together the answer will also be 803.)

As a kicker, hand the spectator a calculator, and ask him to divide the number on line 10 by the number on line 9. In our example, the quotient is ³⁰⁹/₁₉₁ = 1.617. Have the spectator announce the first three digits of the quotient, then show the piece of paper (where you have already written a prediction). He’ll be surprised to see that you’ve already written the number 1.61!

Try this with your own numbers.

So how does this interesting sequence of numbers relate to the golden ratio?  Take any value in the sequence and divide it by the preceding value – what do you get?

For example: ²⁸/₁₇ = 1.647

Try it again for a pair farther down the sequence:

³⁰⁹/₁₉₁ = 1.617

In fact, this manipulation of the Fibonacci series converges to the golden ratio. If you continue the leapfrog process indefinitely, the ratio of consecutive terms gets closer and closer to 1.6180339887 . . . a number with so many amazingly beautiful and mysterious properties that it is often called the golden ratio.