Why don't they teach Newton's calculus of 'What comes next?'

The video begins with an introduction to a familiar mathematical guessing game, where the first few terms suggest a pattern of powers of 2. However, the sequence quickly becomes less predictable. The speaker introduces a method for deciphering such sequences by calculating differences between consecutive terms. This method reveals a hidden pattern of 1s, suggesting a connection to powers of 2. The video sets the stage for exploring the mathematics behind 'what's next' puzzles and introduces the concept of sequence calculus as a way to systematically solve these puzzles.

The speaker explains how to approach sequences using the concept of functions from school calculus, with the original sequence represented as a function f. The introduction of a 0th term leads to the idea of graphing sequences and the concept of differences as slopes of line segments, drawing parallels with derivatives in calculus. The video also introduces the idea of higher differences and the potential for reconstructing sequences from partial information. The speaker then provides examples of differencing simple sequences like powers of 2 and squares, highlighting the similarities and differences from standard calculus.

This paragraph delves into the algebraic patterns found in sequences, particularly focusing on polynomial sequences. The speaker discusses the properties of difference schemes for sequences like squares and cubes, and how these relate to familiar mathematical concepts like factorials. The video also introduces rules for manipulating difference schemes, such as the effect of multiplying a sequence by a constant or adding two sequences together. The speaker emphasizes that all polynomial sequences result in constant rows in their difference schemes, hinting at the nature of the mystery sequence introduced earlier.

The speaker introduces the Gregory-Newton interpolation formula as a powerful tool for solving 'what's next' riddles and finding the underlying polynomial rule of a sequence. The video demonstrates the application of the formula to the mystery sequence from the introduction, showing how it can accurately predict the next terms. The speaker also discusses the broader implications of the formula, including its ability to handle an infinite number of terms and its application to non-integer values, highlighting the beauty and utility of the Gregory-Newton formula in sequence calculus.

The speaker provides a deeper exploration of the Gregory-Newton formula, offering insights into its applications and connections to other mathematical concepts. The video discusses how the formula can be used to solve any 'what's next' puzzle, even with non-traditional sequences like pi, phi, and e. The speaker also touches on the limitations of such puzzles and the importance of scientific context in truly understanding sequences. The video concludes with a promise to provide the mathematical context for the mystery sequence in a future segment.

The speaker shifts focus to the Fibonacci sequence, exploring its difference scheme and how it mirrors the sequence itself. The video highlights the geometric and algebraic properties of the sequence, leading to the formulation of a difference equation. The speaker then connects this to the concept of solving differential equations, showing that the Fibonacci sequence can be solved in a manner similar to linear second-order homogeneous differential equations. The video concludes with a teaser for the proof of the Gregory-Newton formula and its application in real-life numerical calculations.

The speaker provides a sketch of the proof for the Gregory-Newton formula, focusing on sequences that end in a constant row. The video uses the example of n choose 4 to illustrate how the formula works and how it can be extended to other sequences. The speaker also discusses the broader implications of sequence calculus, drawing parallels with school calculus and highlighting the fundamental theorem of sequence calculus. The video concludes with a challenge for the viewer to prove the fundamental theorem themselves and a brief overview of how sequence calculus can be applied to real-life numerical problems.