The binomial theorem (or binomial expansion) is a result of expanding the powers of binomials or sums of two terms. The coefficients of the terms in the expansion are the binomial coefficients . The theorem and its generalizations can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas of mathematics. 

The binomial theorem generalizes special cases which are common and familiar to students of basic algebra: 

(x+y)^1 = x+y 

(x+y)^2 = x^2 + 2xy + y^2 

(x+y)^3 = x^3 + 3x^2y+3xy^2+y^3 

(x+y)^4 = x^4 + 4x^3y + 6x^2y^2+4xy^3+y^4 

 

The binomial theorem also helps explore probability in an organized way: 

Example Question: A friend says that she will flip a coin 5 times. Each time the coin comes up heads, she will give you $10, but each time the coin comes up tails, she gives nothing. What is the probability that you will win $30 playing this game? 

XXX 

Real World Examples of Binomial Theorem 

Distribution of IP addresses 

National economic prediction
 Architecture
 Weather forecasting
 Ranking of candidates 

From <https://www.quora.com/What-are-some-real-world-examples-of-the-use-of-the-binomial-theorem> & <https://brilliant.org/wiki/binomial-theorem-n-choose-k/> 

 

 

 

 

THE NATURAL NUMBERS are the counting numbers:  1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.

 Suppose we have some statement P,n and we want to demonstrate that P,n is true for all n � �. Even if we can provide proofs for P�0, P�1, ..., P�k, where k is some large number, we have accomplished very little. However, there is a general method, the Principle of Mathematical Induction. 

Example Question:-

There is a tower of nn disks, stacked in decreasing size on one of 33 pegs.

How many moves are needed to move all the disks onto a different peg?

Real World Example Of Mathematical Induction:-
From:-
<https://www.quora.com/What-is-the-use-of-Mathematical-Induction-in-real-life>

<https://www.quora.com/What-are-the-real-life-examples-of-the-principle-of-mathematical-induction>

<https://youtu.be/ZpPk99D_Vlg>

The units are m/s, as this is a velocity. We have found the rate of change by looking at the slope.

Clearly, if we were to zoom in closer, our curve would look even more straight and we could get an even better approximation for the slope of the curve. 

  

*https://schools.aglasem.com/46113

*https://www.askiitians.com/revision-notes-new/maths/mathematical-reasoning/

*https://gupea.ub.gu.se/bitstream/2077/38715/1/gupea_2077_38715_1.pdf

*https://www.teachingchannel.org/videos/real-world-math-examples

MATHEMATICAL INDUCTION 

The  Principle of mathematical induction

THE NATURAL NUMBERS are thecounting numbers:  1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.

  Suppose we have some statement P,n and we want to demonstrate that P,n is true for all n � �. Even if we can provide proofs for P�0, P�1, ..., P�k, where k is some large number, we have accomplished very little. However, there is a general method, the Principle of Mathematical Induction.

 Example Question:- 

1. There is a tower of nn disks, stacked in decreasing size on one of 33 pegs.
 
 How many moves are needed to move all the disks onto a different peg?

 
Real World Example Of Mathematical Induction:- 

From:- 

<https://www.quora.com/What-is-the-use-of-Mathematical-Induction-in-real-life>

<https://www.quora.com/What-are-the-real-life-examples-of-the-principle-of-mathematical-induction >

 

 
 <https://youtu.be/ZpPk99D_Vlg> 

Here are a few lists of numbers:

· 21, 16, 11, 6 ...

· 1, 2, 4, 8 ...

Ordered lists of numbers like these are called sequences. Each number in a sequence is called a term.

 | 3, | 5, | 7,… 
 | 1st term | 2nd term | 3rd term

Sequences usually have patterns that allow us to predict what the next term might be.

Series.
Let a_1,a_2,a₃, a_n… be a given sequence. Then, the expression

        a₁ + a₂ + a₃ +,…+ a_n +…

is called the series associated with the given sequence.

Arithmetic progression

A sequence of numbers in which each differs from the preceding one by a constant quantity (e.g. 1, 2, 3, 4, etc.; 9, 7, 5, 3, etc.).

Geometric progression

A progression of numbers with a constant ratio between each number and the one before (e.g. 1, 3, 9, 27, 81).

Given two numbers a and b. We can insert a number A between them so that a,A,b is an Arithmetic Progression. This number A is called the Arithmetic Mean of the two numbers a and b.
      A = (a+b)/2 
The Geometric Mean of two positive numbers a and b is the number root of ab.

CCL:-  HISTORY  
 Evidence is found that Babylonians, some 400 years ago, knew of arithmetic and geometric sequences
 SCIENCE
On plotting straight line graphs for relations between two or more functions/formulas/units, AP can be found and used to find distinct points on the graph without actually using the X and Y axis denotations.

WL:- https://brilliant.org/wiki/geometric-progressions/#applications

https://www.quora.com/What-are-some-examples-of-Arithmetic-Progression-in-Nature

https://matheducators.stackexchange.com/questions/11106/examples-of-arithmetic-and-geometric-sequences-and-series-in-daily-life

https://www.youtube.com/watch?v=AjAO0myllzw

https://www.youtube.com/watch?v=O7x3LBWn-Ao

https://www.youtube.com/watch?v=SjSHVDfXHQ4  (personal favorite)