Download Mathematical Induction (Examples Worksheet) and more Study notes Calculus in PDF only on Docsity! Mathematical Induction (Examples Worksheet) The Method: 1. State the claim you are proving. (Don’t use ghetto P(n) lingo). 2. Write (Base Case) and prove the base case holds for n=a. 3. Write (Induction Hypothesis) say “Assume ___ for some 𝑘 ≥ 𝑎”. 4. Write the WWTS: _________________ 5. Prove the (k+1)th case is true. You MUST at some point use your induction hypothesis in this step. STATE EXACTLY when it’s used. 6. Conclude that the claim holds for all 𝑛 ≥ 𝑎. And QED if deserved! “Some” is a very important word here. MATHEMATICAL INDUCTION PRACTICE Claim: 1 + 3 + 5 + . . . + (2n-1) = n2 We start with the base case. This is usually 0 or 1 if not specified. Start with some examples below to make sure you believe the claim. PROOF: (Base Case): LHS: RHS: (Induction hypothesis): Assume 1+ 3 + . . . + (2k-1) = k2 for some _____. WWTS: Let’s do a geometric proof below: (HINT: draw a 3x3 square composed of 9 balls. Group the balls in L-shapes of size 1, 3, and 5.) Claim: 3n ≤ 3n for all natural numbers n. Start with some examples below to make sure you believe the claim. PROOF: (Base Case): n=1 LHS: 3 RHS: 3 Since 3 ≤ 3, the base case is done. (Inductive hypothesis): Assume that 3k ≤ 3k for some k≥1. WWTS: Then 3(k+1) = = 3k+1. Then, 3n ≤ 3n for all natural numbers n. Notice that the first thing I did was “extract” the 3k from with the 3k+1. Then I substituted using the inductive assumption. Then I substituted a larger quantity for the 3 (which is fine because I am proving a < inequality). Then I added a positive amount on, which allowed me to arrive at the RHS of what I wanted to show. Notice that “working backwards” from the end goal (that is, the 3k+1 in this case) is often helpful when doing proofs by induction on inequalities! Claim: 2+3n < 2n for all n > 3. Start with some examples below to make sure you believe the claim. PROOF: (Base Case): LHS: RHS: (Inductive hypothesis): Assume that WWTS: Claim: 5n-1 is divisible by 4 Start with some examples below to make sure you believe the claim. PROOF: (Base Case): LHS: RHS: (Inductive hypothesis): Assume that WWTS:
