1. Mathematical Induction
Let’s play dominoes!

2. Law of Syllogism
One event makes the next event happen in a logical flow.
Don’t end up in a roadside ditch
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3. Mathematical Induction
Two parts
Basic step: show true for the first term, namely, true for n = 1
Inductive step: Assume true for n = k, and then show true for the next term where
n = k + 1

4. Example 1: Prove 1 + 2 + 3 + . . . + n = 𝑛(𝑛+1) 2
Basic Step S1 = 1(1+1) 2 → is 1 = 1(1+1) 2 ?
TRUE!
Inductive Step
Assume true for n = k
k = 𝑘(𝑘+1) 2
b) Show true for n = k (it works for the next term)
k + (k + 1)= (𝑘+1)(𝑘+1+1) 2
+ (k + 1)= (𝑘+1)(𝑘+2) 2 k
𝑘(𝑘+1) 2
TA – DAH!

5. Example 2: Prove 1 + 2 + 4 + . . . + 2n-1 = 2n - 1
Basic Step S1 = 21 – 1 → is 1 = 21 – 1 ?
TRUE!
Inductive Step
Assume true for n = k
k-1 = 2k - 1
b) Show true for n = k (that it works for the next term)
k (k+1)-1 = 2k+1 - 1
2k − 1
k-1
+ 2k = 2k+1 - 1
TA – DAH!

6. Example 3: Prove 1 + 4 + 9 + . . . + n2 = 𝑛(𝑛+1)(2𝑛+1) 6
Basic Step S1 = 1(1+1)(2·1+1) 6 → is 1 = 1(1+1)(2·1+1) 6 ?
TRUE!
Inductive Step
Assume true for n = k
k2 = 𝑘(𝑘+1)(2𝑘+1) 6
b) Show true for n = k (that it works for the next term)
k2 + (k + 1)2= (𝑘+1)(𝑘+1 +1)(2(𝑘+1)+1) 6
𝑘(𝑘+1)(2𝑘+1) 6 k2
+ (k + 1)2 = (𝑘+1)(𝑘+2)(2𝑘+3) 6
TA – DAH!