Mathematical Induction Honors Algebra II Keeper

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

Example Use mathematical induction to prove that the sum of the first 𝑛 even positive integers is 𝑛 2 +𝑛. That is, prove that 2+4+6+…+2𝑛= 𝑛 2 +𝑛 is true for all positive integers 𝑛.

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

Example 1 3 + 1 15 + 1 35 +…+ 1 2𝑛−1 2𝑛+1 = 𝑛 2𝑛+1

Example 1 2 + 1 2 2 + 1 2 3 +…+ 1 2 𝑛 =1− 1 2 𝑛

Example 1+8+27+…+ 𝑛 3 = 𝑛 2 𝑛+1 2 4

Example 𝑎=1 𝑛 4𝑎−3 =𝑛 2𝑛−1

Example 𝑎=1 𝑛 1 4 𝑎 2 −1 = 𝑛 2𝑛+1

Example 𝑎=1 𝑛 1 𝑎+1 𝑎+2 = 𝑛 2 𝑛+2

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)

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)

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