Mathematical Induction Let’s play dominoes!

Law of Syllogism One event makes the next event happen in a logical flow. https://youtu.be/kIv3m2gMgUU Don’t end up in a roadside ditch (top 5 https://youtu.be/olO8IBM2Ax4)

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

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 1 + 2 + 3 + . . . + k = 𝑘(𝑘+1) 2 b) Show true for n = k + 1 (it works for the next term) 1 + 2 + 3 + . . . + k + (k + 1)= (𝑘+1)(𝑘+1+1) 2 + (k + 1)= (𝑘+1)(𝑘+2) 2 1 + 2 + 3 + . . . + k 𝑘(𝑘+1) 2 TA – DAH!

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 1 + 2 + 4 + . . . + 2k-1 = 2k - 1 b) Show true for n = k + 1 (that it works for the next term) 1 + 2 + 4 + . . . + 2k-1 + 2(k+1)-1 = 2k+1 - 1 2k − 1 1 + 2 + 4 + . . . + 2k-1 + 2k = 2k+1 - 1 TA – DAH!

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 1 + 4 + 9 + . . . + k2 = 𝑘(𝑘+1)(2𝑘+1) 6 b) Show true for n = k + 1 (that it works for the next term) 1 + 4 + 9 + . . . + k2 + (k + 1)2= (𝑘+1)(𝑘+1 +1)(2(𝑘+1)+1) 6 𝑘(𝑘+1)(2𝑘+1) 6 1 + 4 + 9 + . . . + k2 + (k + 1)2 = (𝑘+1)(𝑘+2)(2𝑘+3) 6 TA – DAH!