Prove that the n th Fibonacci number fn is the integer that is closest to the number (1)/(√(5))(

step 1 We were asked to answer questions about the Fibonacci numbers. In part A, we're given the formul

Prove that the n th Fibonacci number fn is the integer that...

Key Concepts

Binet’s Formula

Binet’s formula is a closed-form expression for the Fibonacci numbers that expresses the nth Fibonacci number as a combination of two exponential terms involving the Golden Ratio and its conjugate. This formula provides an analytic tool to study the sequence and facilitates error analysis in approximations.

Error Estimation in Approximation

In the context of Binet’s formula, the term involving the conjugate of the Golden Ratio tends to zero rapidly because its absolute value is less than one. This allows one to show that the main term is very close to an integer, and hence the Fibonacci number is the closest integer to this dominant term. Error estimation here is crucial to justify the accuracy of the approximation.

Fibonacci Sequence

The Fibonacci sequence is a recursively defined sequence where each term is the sum of the two preceding ones. It is widely studied in mathematics due to its prevalence in natural phenomena, combinatorics, and computer science, and serves as a prime example of recursive structures that exhibit exponential growth.

Golden Ratio

The Golden Ratio, often represented by ? (phi), is an irrational number that arises as the positive solution to the equation ?² = ? + 1. Its unique properties make it central to the closed-form analysis of the Fibonacci sequence and many other mathematical structures.

Transcript

00:02 We were asked to answer questions about the fibonacci numbers.

00:07 In part a, we're given the formula for fibonacci numbers in terms of n, and we're asked to show that the n -fibonacci number is the integer closest to 1 over root 5 times 1 plus root 5 over 2 to the n.

00:24 So recall from an example in the text that the n -fibonacci number is 1 over root 5 times 1 plus root 5 over 2 to the n, minus 1 over root 5 times 1 minus root 5 over 2 to the n.

00:52 It should be a plus 1 over root 5 minus.

00:58 So this means in particular that f of n minus 1 over root 5 times 1 plus root 5 over 2 to the n.

01:13 This is an absolute value.

01:17 This is equal to the absolute value of 1 over root 5 times 1 minus root 5 over 2.

01:24 To the n.

01:33 This is equal to, since 1 of a root 5 is positive, 1 over root 5 times the absolute value of 1 minus root 5 over 2 to the nth power.

01:52 And this is the same as 1 over root 5 times the absolute value of 1 minus root 5 over 2 to the nth power.

02:07 And we have that root 5 is going to be greater than root 4, which is 2.

02:18 So it follows root 5 is greater than 1.

02:20 So 1 minus root 5 over 2, 1 minus root 5 is going to be less than.

02:54 So root 5 is going to lie between 2 and the square root of 9, which is 3.

03:07 So it follows that 1 minus root 5 is going to lie between 1 minus 2 is negative 1 and 3.

03:23 1 minus 3 is negative 2.

03:25 And so we have that 1 minus root 5 over 2 is going to lie between negative 1 1 1 half and negative 1.

03:36 And so it follows that the absolute value of 1 minus root 5 over 2 is going to be strictly less than 1 over root 5 times 1 to the end, which is equal to 1 over root 5.

03:55 We have that, again, root 5 is going to be greater than 2, so this is less than 1 half.

04:11 Now, we have that different integers differ by at least 1.

04:32 We have that the distance between fn and whatever root 5 times 1 plus root 5 over 2 to the n is less than 1, which is less than 1.

05:09 That fn is always an integer, and so it follows that fn must be the integer closest to 1 over root 5 times 1 plus root 5 over 2 to the n.

05:42 This is what we wanted to show in part a.

05:47 In part b, we we're asked to find for which n f of n is greater than 1 over root 5 times 1 plus root 5 over 2 to the end, for which n f of n is less than 1 of root 5 times 1 plus root 5 over 2 to the end...

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