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Proofs of the Fibonacci Sequence

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First part is to find an expression in terms of n, the results of the formula:

and prove the expression is correct?

Secondly

A Fibonacci sequence is the basis for a superfast calculation trick as follows:

Turn your back and ask someone to write down any two positive integers (vertically and one below the other). Ask them to add the two and get a third, put the third number below the second and add the last two numbers to get a fourth and so on until the column has ten numbers.

In other words you have 10 numbers of a generalised Fibonacci sequence, each the sum of the preceding two numbers except for the first two that are picked at random. You turn around, draw a line below the last two numbers and immediately write the sum of all 10 numbers. The secret is to multiply the seventh number by 11.

Develop a model for the calculation above, and prove mathematically that the sum of the first 10 numbers in a generalised Fibonacci sequence is always 11 times the seventh number.

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Solution Summary

Step-by-step computations and explanations are shown for both problems. The expert provides proofs of the Fibonacci Sequence.

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Part 1
Recall that the Fibonacci sequences is:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . .

First, try computing a few results using the formula to get an idea of what to expect.
1*2 - 1^2 = 1
1*3 - 2^2 = -1
2*5 - 3^2 = 1
3*8 - 5^2 = -1
5*13 - 8^2 = 1

The results appear to be alternating from 1 to -1.
F(n-1) * F(n+1) - F(n)^2 = (-1)^n

We are ready to make a statement, and to prove the statement by ...

Solution provided by:
J Pye, MSc
Education
  • MSc, California State Polytechnic University, Pomona
  • MBA, University of California, Riverside
  • BSc, California State Polytechnic University, Pomona
  • BSc, California State Polytechnic University, Pomona
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