1. Mathematical Induction
Honors Algebra II
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2. What is mathematical induction
Mathematical induction is used to prove conjectures. More specifically, for this lesson, mathematical induction will be used to prove summation formulas. Steps for induction: Basis Step: Verify that a conjecture 𝑃 𝑛 is true for 𝑛=1 Inductive Hypothesis: Assume that 𝑃 𝑛 is true for 𝑛=𝑘 Inductive Step: Use the assumption to prove that 𝑃 𝑛 is also true for 𝑛=𝑘+1

3. Example
Use mathematical induction to prove that the sum of the first 𝑛 even positive integers is 𝑛 2 +𝑛. That is, prove that …+2𝑛= 𝑛 2 +𝑛 is true for all positive integers 𝑛.

4. Prove using induction
1+3+5+…+ 2𝑛−1 = 𝑛 2

5. Example
…+ 1 2𝑛−1 2𝑛+1 = 𝑛 2𝑛+1

6. Example
… 𝑛 =1− 1 2 𝑛

7. Example
…+ 𝑛 3 = 𝑛 2 𝑛

8. Example
𝑎=1 𝑛 4𝑎−3 =𝑛 2𝑛−1

9. Example
𝑎=1 𝑛 𝑎 2 −1 = 𝑛 2𝑛+1

10. Example
𝑎=1 𝑛 1 𝑎+1 𝑎+2 = 𝑛 2 𝑛+2

11. Find formula for the sum 𝑆 𝑛 of the first 𝑛 terms of each sequence
Find formula for the sum 𝑆 𝑛 of the first 𝑛 terms of each sequence. Then prove your formula using induction
2,6,10,14,…,(4𝑛−2)

12. Find formula for the sum 𝑆 𝑛 of the first 𝑛 terms of each sequence
Find formula for the sum 𝑆 𝑛 of the first 𝑛 terms of each sequence. Then prove your formula using induction
2,7,12,17,…,(5𝑛−3)

13. Find formula for the sum 𝑆 𝑛 of the first 𝑛 terms of each sequence
Find formula for the sum 𝑆 𝑛 of the first 𝑛 terms of each sequence. Then prove your formula using induction
− 1 2 ,− 1 4 ,− 1 8 ,− 1 16 ,…,− 1 2 𝑛