9.4
 In order to prove that a statement, 𝑃𝑘 , is true for all
positive integers, you can use mathematical
induction
 Show the statement is true for k = 1
 Assume that 𝑃𝑘 is true
 Show that if 𝑃𝑘 is true, then 𝑃𝑘+1 must be true
 By induction the statement is true for all positive
integers.
 𝑃𝑘 : 𝑆𝑘 =
𝑘 2 (𝑘+1)2
4
 𝑃𝑘 : 𝑆𝑘 = 1 + 5 + 9 + ⋯ + 4 𝑘 − 1 − 3 + (4k − 3)
 𝑃𝑘 : 𝑘 + 3 < 5𝑘 2
 𝑃𝑘 : 3𝑘 > 2𝑘 + 1
 Use induction to prove the following formula:
 𝑆𝑛 = 1 + 3 + 5 + 7 + ⋯ + 2𝑛 − 1 = 𝑛2
 Use induction to prove the following true
2
2
2
2
 𝑆𝑛 = 1 + 2 + 3 + ⋯ + 𝑛 =
𝑛(𝑛+1)(2𝑛+1)
6
 Pg 642 #5-17 odd