Problems Plus 24: Explicit Formula for the n-th Fibonacci Number 🤯

In this video I go over an in-depth derivation to determine an explicit formula for the n-th Fibonacci number. The Fibonacci sequence is the sequence of terms where the previous 2 numbers are added together. Prior to making this video, I had not known there was an explicit formula to determine any given number in the Fibonacci sequence. The derivation involves first determining the Maclaurin series for the given function by writing it out as a power series. Doing so yield the coefficients of the Maclaurin series to be just the Fibonacci numbers. The next part of the derivation is to solve for the series of the given function again but this time using a different method, by using partial fractions. This yields two partial fractions that are in the form of the sum of a convergent geometric series. Replacing the partial fractions with their corresponding Geometric series, simplifying the result, and comparing with our prior Maclaurin series, I note that we have in fact an explicit formula for the n-th Fibonacci number. Absolutely amazing and mind-boggling stuff!

The timestamps of key parts of the video are listed below:

- Problem 24: Series involving Fibonacci series: 0:00
- Solution to (a): The function as a Maclaurin series: 1:37
- Comparing coefficients of powers of x: 7:25
- Each coefficient is equal to the n-th Fibonacci number: 11:27
- Solution to (b): Explicit formula for the n-th Fibonacci number: 13:06
- Completing the square: 13:32
- Writing f(x) as partial fractions: 21:41
- Summary of f(x) as partial fractions: 29:13
- Partial fractions are in the form of the sum of a convergent geometric series: 35:15
- Simplifying the resulting series: 43:52
- Explicit formula for the n-th Fibonacci number: 49:24

This video was taken from my earlier video listed below:

- Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P

Related Videos:

Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .

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