Mathematical Induction Section 9.4 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 11/27/2018 7:19 AM 9.4: Mathematical Induction

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. 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Review What is the pattern and the equation? Just because we came up with the equation, will it always work??? 11/27/2018 7:19 AM 9.4: Mathematical Induction

Let’s Revisit Geometry There are four circles here. How many regions can we make with each circle? 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 1 Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟐+𝟒+𝟔+𝟖+…+𝟐𝒏= 𝒏 𝟐 +𝒏 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 1 Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟐+𝟒+𝟔+𝟖+…+𝟐𝒏= 𝒏 𝟐 +𝒏 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 2 Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟏 +𝟑+𝟓+𝟕+…+ 𝟐𝒏−𝟏 = 𝒏 𝟐 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 2 Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟏 +𝟑+𝟓+𝟕+…+ 𝟐𝒏−𝟏 = 𝒏 𝟐 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 2 Use mathematical induction to prove the following formula: 𝑺 𝒏 =𝟏 +𝟑+𝟓+𝟕+…+ 𝟐𝒏−𝟏 = 𝒏 𝟐 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 . 11/27/2018 7:19 AM 9.4: Mathematical Induction

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 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 4 Find the sum of 𝒊=𝟏 𝟕 𝒊 𝟑 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 5 Find the sum of 𝒊=𝟏 𝟒 𝟔𝒊−𝟒 𝒊 𝟐 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Your Turn Find the sum of 𝒊=𝟏 𝟒 𝟐 𝒊 𝟐 +𝟑 𝒊 𝟑 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Example 6 Given the image below, how many oranges are in this square pyramid that is 10 layers high? 11/27/2018 7:19 AM 9.4: Mathematical Induction

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. 11/27/2018 7:19 AM 9.4: Mathematical Induction

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. 11/27/2018 7:19 AM 9.4: Mathematical Induction

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. 11/27/2018 7:19 AM 9.4: Mathematical Induction

9.4: Mathematical Induction Assignment Worksheet 11/27/2018 7:19 AM 9.4: Mathematical Induction