Proof by Induction

Starter: write expressions for FM proof by induction: KUS objectives BAT understand and use the principle of proof by induction BAT prove results about divisibility Starter: write expressions for 1 𝑛 𝑟 = 𝑛 2 (𝑛+1) 1 𝑛+1 𝑟 2 = 1 6 (𝑛+1)(𝑛+2)(2𝑛+3) 11 𝑛+1 𝑟 = 1 2 𝑛+1 𝑛+2 −30

Proof by mathematical Induction Sigma notation notes Proof by mathematical Induction 1. Basis Prove the statement is true for n = 1 2. Assumption That the statement is true for n = k 3. Inductive Show that the statement is then true for n = k + 1 4. Conclusion That the statement is then true for all positive integers, n

Calculate the matrices and WB C1 𝑎) 1 2 0 1 𝑛 = 1 2𝑛 0 1 𝑎) 1 2 0 1 𝑛 = 1 2𝑛 0 1 Let A= 1 1 0 1 Calculate the matrices and Make a conjecture about the matrix An and prove it by mathematical induction Does it matter if we multiply the matrix at the front instead of the back - Try it and see (solution is the same) Hypothesis:

Conjecture’; Prove by induction that WB C1 (cont) Conjecture’; Prove by induction that True for n = 1 LHS RHS Assume Does it matter if we multiply the matrix at the front instead of the back - Try it and see (solution is the same) True for n = k+1 True for all integers

n = 1 True for n = 1 Assume n = k True for n = k+1 WB C2 Prove by induction that for all positive integers n, 1 −1 0 2 𝑛 = 1 1− 2 𝑛 0 2 𝑛 𝐿𝐻𝑆 1 −1 0 2 𝑛 = 1 −1 0 2 𝑅𝐻𝑆 1 1− 2 𝑛 0 2 𝑛 = 1 −1 0 2 n = 1 True for n = 1 Assume 1 −1 0 2 𝑘 = 1 1− 2 𝑘 0 2 𝑘 n = k True for n = k+1 1 −1 0 2 𝑘+1 = True for n = 1, true for all n. true for n = k + 1 if true for n = k.

n = 1 True for n = 1 Assume n = k True for n = k+1 WB C3 Prove by induction that for all positive integers n, −2 9 −1 4 𝑛 = −3𝑛+1 9𝑛 −𝑛 3𝑛+1 𝐿𝐻𝑆 −2 9 −1 4 1 = −2 9 −1 4 𝑅𝐻𝑆 −3𝑛+1 9𝑛 −𝑛 3𝑛+1 = −2 9 −1 4 n = 1 True for n = 1 Assume −2 9 −1 4 𝑘 = −3𝑘+1 9𝑘 −𝑘 3𝑘+1 n = k −2 9 −1 4 𝑘+1 = −2 9 −1 4 𝑘 −2 9 −1 4 = −3𝑘+1 9𝑘 −𝑘 3𝑘+1 −2 9 −1 4 = −3𝑘−2 9𝑘+9 −𝑘−1 3𝑘+4 = −3 𝑘+1 +1 9 𝑘+1 − 𝑘+1 3 𝑘+1 +1 True for n = k+1 true for n = k + 1 if true for n = k. True for n = 1, true for all n.

WB C4 Prove by mathematical induction the following statements 𝑎) 1 2 0 1 𝑛 = 1 2𝑛 0 1 𝑏) 3 −4 1 −1 𝑛 = 2𝑛+1 −4𝑛 𝑛 −2𝑛+1 𝑐) 2 0 1 1 𝑛 = 2 𝑛 0 2 𝑛 −1 1 𝑑) 5 −8 2 −3 𝑛 = 4𝑛+1 −8𝑛 2𝑛 1−4𝑛 𝑒) 2 5 0 1 𝑛 = 2 𝑛 5 2 𝑛 −1 0 1 Does it matter if we multiply the matrix at the front instead of the back - Try it and see (if solution is the same)

Crucial points 1. Try to really understand the principle behind proof by induction Many students find this difficult, and it can take time for the ideas to really sink in. The Notes and Examples should help. 2. Always think about what you are aiming for When you take the assumed result for n = k and add on the (k + 1)th term, you want to rearrange this to get the formula for n = k + 1. It may help to actually write down the result you are looking for. 3. Be careful with algebraic manipulation In many questions on this topic, the algebraic manipulation required can appear daunting. It is vital that wherever possible you take out common factors. Never expand two or more terms and then try to factorise the result. See Example 1 in the Notes and Examples. 4. Make sure that you write out the proof correctly If you are proving a result using proof by induction, the tidying up of the result to prove the result for n = 1, or n = 2, as appropriate, is a vital part of the proof and

self-assess One thing learned is – One thing to improve is –

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Notes