18 1.3 The Natural Numbers and Mathematical Induction Exercises 1.3.1 Prove the following using Mathematical Induction. (a) 12 +22 +· · ·+n2 =n(n+1)(2n+1) 6 for all n∈N. (b) 13 +23 +· · ·+n3 =n 2(n+1)2 4 for all n∈N. (c) 1+3+· · ·+(2n−1) =n2 for all n∈N. 1.3.2 Prove the following using Mathematical Induction. (a) 9n −5n is divisible by 4 for all n∈N. (b) 7n −1 is divisible by 3 for all n∈N. (c) 32n −1 is divisible by 8 for all n∈N. (d) xn −yn is divisible by x−y for all n∈Nwhere x,y ∈Z, x̸ =y. 1.3.3 Prove the following using Mathematical Induction. (a) 1+3n≤4n for all n∈N. (b) 1+2n≤2n for all n∈N, n≥3. (c) n2 ≤3n for all n∈N. (d) n3 ≤3n for all n∈N. (Hint: Check the cases n=1 andn=2 directly and then use induction for n≥3.) 1.3.4 Given a real number a̸=1, prove that 1+a+a2 +· · ·+an = 1−an+1 1−a for all n∈N. 1.3.5 ▶The Fibonacci sequence is defined by a1 =a2 =1 and an+2 =an+1 +an for n≥1. Prove that an = 1 √5h 1+√5 2 n − 1− √5 2 ni. 1.3.6 Let a≥ −1. Prove by induction that (1+a)n ≥1+na for all n∈N. 1.3.7 ▷Let a,b∈Randn∈N. Use Mathematical Induction to prove the binomial theorem (a+b)n = n ∑ k=0 n k akbn−k, where n k = n! k!(n−k)! .
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