# Category:Proofs by Induction

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## Subcategories

This category has the following 3 subcategories, out of 3 total.

### L

## Pages in category "Proofs by Induction"

The following 106 pages are in this category, out of 106 total.

### A

### C

- Cardinality of Proper Subset of Finite Set
- Cardinality of Set of Injections/Formal Proof
- Cauchy's Group Theorem/Proof 1
- Cauchy's Mean Theorem/Proof 1
- Closed Form for Triangular Numbers/Proof by Induction
- Coefficients of Polynomial Product
- Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices
- Congruence Modulo Power of p as Linear Combination of Congruences Modulo p
- Consecutive Fibonacci Numbers are Coprime
- Convex Hull is Smallest Convex Set containing Set/Lemma
- Cyclicity Condition for Units of Ring of Integers Modulo n

### D

- De Moivre's Formula/Positive Integer Index/Proof 1
- Divisibility of Fibonacci Number
- Divisor of Integer/Examples/6 divides 7^n - 1
- Divisor of Integer/Examples/6 divides 7^n - 1/Proof 1
- Divisor of Integer/Examples/63 divides 8^2n - 1
- Divisor of Integer/Examples/63 divides 8^2n - 1/Proof 1
- Divisor of Integer/Examples/8 divides 3^2n + 7/Proof 1
- Divisor of Integer/Examples/80 divides 9^2n - 1/Proof 1

### E

### F

- Fermat's Little Theorem/Proof 4
- Fibonacci Number as Sum of Binomial Coefficients
- Fibonacci Number with Negative Index
- Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements/Proof 1
- Finite Union of Sets in Additive Function
- Finite Union of Sets in Subadditive Function
- Fundamental Theorem of Finite Abelian Groups

### I

### M

### N

- Natural Number Addition is Commutative/Proof 2
- Natural Number Multiplication Distributes over Addition/Proof 2
- Natural Number Multiplication is Associative/Proof 2
- Negation of Ordering of Natural Numbers is Provable
- Niven's Theorem/Lemma
- Nonzero Natural Number is Successor
- Nth Derivative of Exponential of Minus One over x
- Nth Derivative of Natural Logarithm

### O

### P

- Polynomial is of Exponential Order Epsilon
- Power Function is Strictly Increasing on Positive Elements
- Power of Identity is Identity
- Powers of 16 Modulo 20/Proof 1
- Prime Power of Sum Modulo Prime
- Principle of Finite Choice
- Product of Sums of Four Squares/Corollary
- Proper Subset of Finite Ordinal is Equivalent to Smaller Ordinal
- Properties of Product of Identity plus Operator Raised to Powers of 2

### R

### S

- Sine of X over X as Infinite Product
- Size of Complete Graph
- Size of Linearly Independent Subset is at Most Size of Finite Generator/Proof 1
- Strictly Increasing Sequence of Natural Numbers
- Sum of Even Index Binomial Coefficients
- Sum of Even Index Binomial Coefficients/Proof 2
- Sum of Geometric Sequence/Proof 1
- Sum of Integrals on Adjacent Intervals for Integrable Functions/Corollary
- Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers
- Sum of Powers of 2/Proof 2
- Sum of Sequence of Cubes/Proof by Induction
- Sum of Sequence of Even Index Fibonacci Numbers
- Sum of Sequence of Fibonacci Numbers
- Sum of Sequence of Odd Cubes
- Sum of Sequence of Odd Index Fibonacci Numbers
- Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient/Proof 1
- Sum of Sequence of Products of 3 Consecutive Integers
- Sum of Sequence of Products of Consecutive Integers/Proof 1
- Sum of Sequence of Products of Consecutive Odd and Consecutive Even Numbers
- Sum of Sequence of Products of Consecutive Reciprocals/Proof 1
- Sum of Sequence of Squares of Fibonacci Numbers
- Sum of Sequence of Squares/Proof by Induction
- Summation Formula for Polygonal Numbers
- Summation is Linear/Scaling of Summations
- Summation is Linear/Sum of Summations