Reference no: EM133548720
Question: The Fibonacci sequence of numbers is defined as follows: The first and second numbers are both 1. After that, each number in the sequence is the sum of the two preceding numbers. Thus, the Fibonacci sequence is as follows:
1, 1, 2, 3, 5, 8, 13, 21, ...
If F(n) stands for the nth value in the sequence, then this definition can be expressed as
F(1) = 1
F(2) = 1
F(n) = F(n - 1) + F(n - 2) for n > 2
The value of F at position n is defined using the value of F at two smaller positions. Something that is defined in terms of "smaller versions" of itself is said to be recursive, so the Fibonacci sequence is recursive. Using the definition of the Fibonacci sequence, compute the value of F(20).
A formula for F(n) is
Using the formula (and a calculator), compute the value of F(20).
What is your opinion on the relative clarity, elegance, and efficiency of the two algorithms (using the definition and using the formula) to compute F(20)? Would your answer change if you considered F(100)?
If you have access to a spreadsheet, program it to compute successive values of the Fibonacci sequence. (Hint: Enter 1 in cells A1 and B1, and enter = A1 + B1 in cell C1. Continue along row 1 with a similar formula.) Use this method to compute F(20).
Elsewhere on the same spreadsheet, use one cell to enter the value of n, and another cell to enter the formula from part (b) above. (Hint: you will need the squareroot function and the power function.) Use this method to compute F(20).
On your spreadsheet, use both methods to compute F(50). Which appears to be more efficient?
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