Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Mathematical Induction Proof Technique

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, ^{n}Σ_{k=1} = [n(n+1)]/2

(a) Prove, by induction, that, for every integer ≥ 5, 2^{n} > n^{2}.

(b) Prove, by induction, that any integer n ≥ 2 can be expressed as a finite product of primes.

**The strings**:
S_{7}P_{2}A_{21} (Identity - Physical Property).
**The math**:

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 2^{5} = 32 and 5^{2} = 25

So, 2^{5} > 5^{2}

Assume 2^{n} > n^{2}

We need to show that 2^{n+1} > (n+1)^{2}

2(2^{n}) > 2(n)^{2}

We need to show that 2(n)^{2} >(n+1)^{2} for n > 5.

2(n)^{2} - (n^{2} +2n -1) = (n-1)^{2}

(n+1)^{2} - (n^{2} +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.

Math

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.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

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.

Periodic Table

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