Easy Steps to Learn Proof by Induction for Natural Numbers

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Ann

30/05/2023

Further Maths

Chapter 8: Proof by induction

Subjects

Further Maths

Further maths pure

Proof

Easy Steps to Learn Proof by Induction for Natural Numbers

Mathematical Induction and Proof Methods - A comprehensive guide covering proof by induction for natural numbers, examples of summation proofs using induction, and divisibility proofs with induction for integers.

Introduces three main types of proofs: summation, divisibility, and matrix proofs
Details the four essential steps of the induction process: basis, assumption, inductive step, and conclusion
Provides practical examples of each proof type with detailed solutions
Covers advanced concepts including matrix operations and complex summation formulas
Emphasizes the importance of clear mathematical reasoning and systematic proof construction

30/05/2023

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8- Preet by induction
we can use proof by induction whenever we want to
show some properly holds for all integers (usually positive
up to

Page 2: Summation Proofs

This page demonstrates a detailed example of summation proofs using induction to prove that the sum of $2r-1$ equals n² for all positive integers.

Example: The proof shows how to verify that Σ $2r-1$ = n² from r=1 to n

Highlight: The solution follows the standard induction steps:

Base case verification for n=1
Assumption for n=k
Inductive step proving n=k+1
Final conclusion

Definition: The left-hand side $LHS$ represents the summation while the right-hand side $RHS$ represents the simplified form.

View

Page 3: Advanced Summation Proofs

This page explores a more complex summation proof involving cubic terms and demonstrates how to prove that the sum of cubes equals a quarter of n² $n+1$ ².

Example: Proves that Σr³ = ¼n² $n+1$ ² for r=1 to n

Highlight: The solution requires careful algebraic manipulation and understanding of polynomial expressions.

View

Page 4: Divisibility Proofs - Method 1

This page introduces divisibility proofs using induction, specifically proving that 3²ⁿ+11 is divisible by 4 for all positive integers.

Definition: A divisibility proof shows that one expression is always divisible by another number.

Example: Proves 3²ⁿ+11 is divisible by 4 using the first method.

Highlight: The proof demonstrates how to handle exponential expressions in induction.

View

Page 5: Divisibility Proofs - Method 2

This page presents an alternative approach to the divisibility proof from page 4 and introduces a new example involving powers of 8 and 3.

Example: Shows how 8ⁿ-3ⁿ is divisible by 5 for all positive integers n.

Highlight: The second method often provides a more elegant solution by focusing on the difference between consecutive terms.

View

Page 6: Matrix Proofs - Part 1

This page introduces matrix proofs using induction, showing how to prove properties of matrix powers.

Definition: Matrix proofs involve proving statements about matrices raised to different powers.

Example: Proves a specific matrix equality using induction.

Highlight: Matrix proofs require understanding of matrix multiplication and properties.

View

Page 7: Matrix Proofs - Continuation

This page continues the matrix proof example, completing the inductive step and conclusion.

Highlight: The solution demonstrates careful matrix multiplication and algebraic manipulation.

Example: Shows the completion of the matrix proof from page 6.

View

Page 8: Advanced Matrix Proofs

This page presents another matrix proof example with more complex matrices and relationships.

Example: Demonstrates a proof involving 2x2 matrices with specific patterns.

Highlight: The solution requires careful attention to matrix multiplication rules and pattern recognition.

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Subjects

Further Maths

Further maths pure

Proof

Further Maths

•

30 May 2023

•

8 pages

Easy Steps to Learn Proof by Induction for Natural Numbers

Ann

@ann_jznv

Introduces three main types of proofs: summation, divisibility, and matrix proofs... Show more

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Page 2: Summation Proofs

This page demonstrates a detailed example of summation proofs using induction to prove that the sum of $2r-1$ equals n² for all positive integers.

Example: The proof shows how to verify that Σ $2r-1$ = n² from r=1 to n

Highlight: The solution follows the standard induction steps:

Base case verification for n=1
Assumption for n=k
Inductive step proving n=k+1
Final conclusion

Definition: The left-hand side $LHS$ represents the summation while the right-hand side $RHS$ represents the simplified form.

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Page 3: Advanced Summation Proofs

This page explores a more complex summation proof involving cubic terms and demonstrates how to prove that the sum of cubes equals a quarter of n² $n+1$ ².

Example: Proves that Σr³ = ¼n² $n+1$ ² for r=1 to n

Highlight: The solution requires careful algebraic manipulation and understanding of polynomial expressions.

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Page 4: Divisibility Proofs - Method 1

This page introduces divisibility proofs using induction, specifically proving that 3²ⁿ+11 is divisible by 4 for all positive integers.

Definition: A divisibility proof shows that one expression is always divisible by another number.

Example: Proves 3²ⁿ+11 is divisible by 4 using the first method.

Highlight: The proof demonstrates how to handle exponential expressions in induction.

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Page 5: Divisibility Proofs - Method 2

This page presents an alternative approach to the divisibility proof from page 4 and introduces a new example involving powers of 8 and 3.

Example: Shows how 8ⁿ-3ⁿ is divisible by 5 for all positive integers n.

Highlight: The second method often provides a more elegant solution by focusing on the difference between consecutive terms.

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Page 6: Matrix Proofs - Part 1

This page introduces matrix proofs using induction, showing how to prove properties of matrix powers.

Definition: Matrix proofs involve proving statements about matrices raised to different powers.

Example: Proves a specific matrix equality using induction.

Highlight: Matrix proofs require understanding of matrix multiplication and properties.

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Page 7: Matrix Proofs - Continuation

This page continues the matrix proof example, completing the inductive step and conclusion.

Highlight: The solution demonstrates careful matrix multiplication and algebraic manipulation.

Example: Shows the completion of the matrix proof from page 6.

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Page 8: Advanced Matrix Proofs

This page presents another matrix proof example with more complex matrices and relationships.

Example: Demonstrates a proof involving 2x2 matrices with specific patterns.

Highlight: The solution requires careful attention to matrix multiplication rules and pattern recognition.

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Page 1: Introduction to Proof by Induction

This page introduces the fundamental concepts of mathematical induction and its applications. The content explains how induction can be used to prove properties for all integers, particularly focusing on positive integers up to infinity.

Definition: Proof by induction is a mathematical method used to prove statements true for all natural numbers.

Highlight: The four essential steps of the induction process are:

Basis Step - Prove for n=1
Assumption Step - Assume true for n=k
Inductive Step - Prove for n=k+1
Conclusion Step - Conclude true for all positive integers

Vocabulary: Z represents the set of all integers, while N represents the set of natural $positive$ numbers.

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