18 1.3 The Natural Numbers and Mathematical Induction Exercises 1.3.1 Prove the following using Mathematical Induction. + 22 2 n(n+1)(2n+1) (a) 12 + ··· + n = for all n ∈ N. 6 2(n+1)2 + 23 3 n (b) 13 + ··· + n = for all n ∈ N. 4 (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 ∈ N where x,y ∈ Z, x ̸ = 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 and n = 2 directly and then use induction for n ≥ 3.) 1.3.4 Given a real number a ̸ = 1, prove that 2 n 1 − a n+1 1+ a + a + ··· + a = for all n ∈ N. 1− a 1.3.5 ▶ The Fibonacci sequence is defned by a1 = a2 = 1 and an+2 = an+1 + an for n ≥ 1. Prove that √ √ 1 h 1 + 5 n 1− 5 ni an = √ − . 5 2 2 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 ∈ R and n ∈ N. Use Mathematical Induction to prove the binomial theorem n n (a + b)n kbn−k = ∑ k a , k=0 n n! where = . k k!(n− k)!
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