Mathematical Induction Proof Technique
Strings (SiPjAjk) = S7P2A21 Base Sequence = 12735 String Sequence = 12735 - 2 - 21
Mathematical induction proof technique is well suited for proof problems of the type:
For a given population of integers, some event occurs. An example of this type of proof problem is as follows:
For all integers n ≥ 1, nΣk=1 = [n(n+1)]/2
(a) Prove, by induction, that, for every integer ≥ 5, 2n > n2.
(b) Prove, by induction, that any integer n ≥ 2 can be expressed as a finite product of primes.
S7P2A21 (Identity - Physical Property).
Pj Problem of Interest is of type identity (physical property). Proofs establish truths. So they are identity problems.
Figure 121.2 illustrates the steps for proving by induction:
(1) Establish the truth of the statement for n = 1
(2) Assume the statement is true for n
(3) Establish the truth for n + 1.
(a) In this problem we have 5 as the lower bound instead of 1
So, for n=5, we have 25 = 32 and 52 = 25
So, 25 > 52
Assume 2n > n2
We need to show that 2n+1 > (n+1)2
2(2n) > 2(n)2
We need to show that 2(n)2 >(n+1)2 for n > 5.
2(n)2 - (n2 +2n -1) = (n-1)2
(n+1)2 - (n2 +2n -1) = 2
So, for n > 5, 2(n)2 > (n+1)2.
(b) Statement is true for n = 2.
Assume statement is true for all integers j, where 2 < j < n.
If n + 1 is prime, then statement is proved
If n + 1 is not prime, then it has a prime divisor, p.
So, there is an integer q, where 2 < q < n such that (n + 1) = pq.
But q can be expressed as a finite product of primes
So, n + 1 can be expressed as a finite product of primes.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
Single Variable Functions
Ordinary Differential Equations (ODEs)
Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation
Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions
Derivation Of Heat Equation For A One-Dimensional Heat Flow
The Universe is composed of matter and radiant energy. Matter is any kind of mass-energy that moves with velocities less than the velocity of light. Radiant energy is any kind of mass-energy that moves with the velocity of light.
Composition And Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Molecular Structure Of Matter
Molecular Shapes: Bond Length, Bond Angle
Molecular Shapes: Valence Shell Electron Pair Repulsion
Molecular Shapes: Orbital Hybridization
Molecular Shapes: Sigma Bonds Pi Bonds
Molecular Shapes: Non ABn Molecules
Molecular Orbital Theory
More Pj Problem Strings