1. Mathematical Induction
Section 9.4
Precalculus PreAP/Dual, Revised ©2017
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9.4: Mathematical Induction

2. 9.4: Mathematical Induction
Definitions
Mathematical induction is a form of mathematical proof and it is deductive reasoning
Just because a rule, pattern, or formula seems to work for several values of 𝒏, you cannot simply decide that it is valid for all values of 𝒏 without going through a legitimate proof.
The Principle of Mathematical Induction: Let 𝑺 𝒏 be a statement involving the positive integer 𝒏. If
1. 𝑺 𝒏 is true, and
2. the truth of 𝑺 𝒌 implies the truth of 𝑺 𝒌+𝟏 , for every positive integer 𝒌, then 𝑺 𝒏 must be true for all integers 𝒏.
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9.4: Mathematical Induction

3. 9.4: Mathematical Induction
Review
What is the pattern and the equation? Just because we came up with the equation, will it always work???
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9.4: Mathematical Induction

4. Let’s Revisit Geometry
There are four circles here. How many regions can we make with each circle?
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9.4: Mathematical Induction

5. 9.4: Mathematical Induction
Steps
Prove that it works for one case
Prove that if the formula works for some number 𝒌 (any integer), then it also works for 𝒌+𝟏
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9.4: Mathematical Induction

6. 9.4: Mathematical Induction
Example 1
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟐+𝟒+𝟔+𝟖+…+𝟐𝒏= 𝒏 𝟐 +𝒏 .
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9.4: Mathematical Induction

7. 9.4: Mathematical Induction
Example 1
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟐+𝟒+𝟔+𝟖+…+𝟐𝒏= 𝒏 𝟐 +𝒏 .
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9.4: Mathematical Induction

8. 9.4: Mathematical Induction
Example 2
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟏 +𝟑+𝟓+𝟕+…+ 𝟐𝒏−𝟏 = 𝒏 𝟐 .
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9.4: Mathematical Induction

9. 9.4: Mathematical Induction
Example 2
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟏 +𝟑+𝟓+𝟕+…+ 𝟐𝒏−𝟏 = 𝒏 𝟐 .
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9.4: Mathematical Induction

10. 9.4: Mathematical Induction
Example 2
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟏 +𝟑+𝟓+𝟕+…+ 𝟐𝒏−𝟏 = 𝒏 𝟐 .
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9.4: Mathematical Induction

11. 9.4: Mathematical Induction
Example 3
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟔 +𝟏𝟏+𝟏𝟔+𝟐𝟏+…+ 𝟓𝒏+𝟏 = 𝒏(𝟓𝒏+𝟕) 𝟐 .
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9.4: Mathematical Induction

12. 9.4: Mathematical Induction
Example 3
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟔 +𝟏𝟏+𝟏𝟔+𝟐𝟏+…+ 𝟓𝒏+𝟏 = 𝒏(𝟓𝒏+𝟕) 𝟐 .
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9.4: Mathematical Induction

13. 9.4: Mathematical Induction
Example 3
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟔 +𝟏𝟏+𝟏𝟔+𝟐𝟏+…+ 𝟓𝒏+𝟏 = 𝒏(𝟓𝒏+𝟕) 𝟐 .
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9.4: Mathematical Induction

14. 9.4: Mathematical Induction
Example 3
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟔 +𝟏𝟏+𝟏𝟔+𝟐𝟏+…+ 𝟓𝒏+𝟏 = 𝒏(𝟓𝒏+𝟕) 𝟐 .
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9.4: Mathematical Induction

15. 9.4: Mathematical Induction
Your Turn
Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟑+𝟔+𝟗+𝟏𝟐+…+𝟑𝒏= 𝟑 𝟐 𝒏 𝒏+𝟏 .
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9.4: Mathematical Induction

16. Sum of Powers of Integers
𝟏+𝟐+𝟑+𝟒+…+𝒏= 𝒏 𝒏+𝟏 𝟐
𝟏 𝟐 + 𝟐 𝟐 + 𝟑 𝟐 + 𝟒 𝟐 +…+ 𝒏 𝟐 = 𝒏 𝒏+𝟏 𝟐𝒏+𝟏 𝟔
𝟏 𝟑 + 𝟐 𝟑 + 𝟑 𝟑 + 𝟒 𝟑 +…+ 𝒏 𝟑 = 𝒏 𝟐 𝒏+𝟏 𝟐 𝟒
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9.4: Mathematical Induction

17. 9.4: Mathematical Induction
Example 4
Find the sum of 𝒊=𝟏 𝟕 𝒊 𝟑
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9.4: Mathematical Induction

18. 9.4: Mathematical Induction
Example 5
Find the sum of 𝒊=𝟏 𝟒 𝟔𝒊−𝟒 𝒊 𝟐
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9.4: Mathematical Induction

19. 9.4: Mathematical Induction
Your Turn
Find the sum of 𝒊=𝟏 𝟒 𝟐 𝒊 𝟐 +𝟑 𝒊 𝟑
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9.4: Mathematical Induction

20. 9.4: Mathematical Induction
Example 6
Given the image below, how many oranges are in this square pyramid that is 10 layers high?
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9.4: Mathematical Induction

21. 9.4: Mathematical Induction
Example 7
A clock at London’s Big Ben chimes every half hour. At the beginning of the n’th hour, the clock chimes n times. (For example, at 8:00 am the clock chimes 8 times, at 2:00 pm the clock chimes fourteen times, and at midnight the clock chimes 24 times.) The clock also chimes once at half-past every hour. Determine how many times in total the clock chimes in one full day. Use sigma notation to write the form of the series before finding its sum.
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9.4: Mathematical Induction

22. 9.4: Mathematical Induction
Example 7
A clock at London’s Big Ben chimes every half hour. At the beginning of the n’th hour, the clock chimes n times. (For example, at 8:00 am the clock chimes 8 times, at 2:00 pm the clock chimes fourteen times, and at midnight the clock chimes 24 times.) The clock also chimes once at half-past every hour. Determine how many times in total the clock chimes in one full day. Use sigma notation to write the form of the series before finding its sum.
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9.4: Mathematical Induction

23. 9.4: Mathematical Induction
Example 7
A clock at London’s Big Ben chimes every half hour. At the beginning of the n’th hour, the clock chimes n times. (For example, at 8:00 am the clock chimes 8 times, at 2:00 pm the clock chimes fourteen times, and at midnight the clock chimes 24 times.) The clock also chimes once at half-past every hour. Determine how many times in total the clock chimes in one full day. Use sigma notation to write the form of the series before finding its sum.
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9.4: Mathematical Induction

24. 9.4: Mathematical Induction
Assignment
Worksheet
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9.4: Mathematical Induction