11.4 Homework (handwritten):
11.4a: p. 1063: #11, 13, 15
11.4b: p. 1063: #17, 19, 21
11.4c: p. 1063: #23, 25, 27
11.4: Mathematical Induction
S’pose you wanted to prove the following statement is true for all
natural numbers n:
𝑛
∑𝑖 = 1+ 2+3 +⋯+ 𝑛 =
𝑖=1
𝑛(𝑛 + 1)
2
The Principle of Mathematical Induction is a method of proof that can
be used to show a statement is true for every natural number. Every
induction proof must have 5 steps:
1)
2)
3)
4)
Define the statement you intend to prove is true.
Show the statement is true for one or more base cases.
Assume the statement is true for 𝑛 = 𝑘.
Show that your assumption implies the statement is true for its
successor, 𝑛 = 𝑘 + 1. This usually requires some “algebra magic”
5) Write a conclusion.
*Always work on only one side of the equation!
Use The Principle of Mathematical Induction (PMI) to show that each of
the given statements are true for all natural numbers.
1.
1+ 2+ 3+⋯+𝑛 =
𝑛(𝑛+1)
2
2.
4 + 7 + 10 … + (3𝑛 + 1) =
𝑛(3𝑛+5)
2
3.
12 + 32 + 52 + ⋯ + (2𝑛 − 1)2 =
𝑛(2𝑛+1)(2𝑛−1)
3
4.
𝑛−1
1 + 4 + 16 + ⋯ + 4
=
4𝑛 −1
3
5.
3 is a factor of 𝑛3 + 2𝑛.
6.
5 is a factor of 6𝑛 − 1.
7.
4 is a factor of 𝑛2 (𝑛 + 1)2