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1. Prove by induction that 1 · 2 + 2 · 3 + 3 · 4 + · · · + n(n + 1) = n(n + 1)(n + 2) 3 2. Prove by induction that for all positive integers n 6|2n 3 + 3n 2 + n. 3. Prove by induction that for all positive integers n that Xn i=1 i(i!) = (n + 1)! − 1, 4. Find an expression for 1 − 3 + 5 − 7 + 9 − 11 + · · · + (−1)n−1 (2n − 1) and prove using induction that your expression is correct. 5. Let {f0, f1, f2, . . . } be the Fibonacci numbers. So f0 = 1 and f1 = 1 and fi = fi−1 + fi−2. (So f2 = 2, f3 = 3 and f4 = 5 and so on.) (a) Prove by induction that f0 + f1 + · · · + fk = fk+2 − 1. (b) Prove by induction that Xn i=0 f 2 i = fnfn+1. (c) Prove by strong induction that fn+5 ≡ 3fn (mod 5) 6. Consider a sequence {x1, x2, x3, x4, x5, ....} where x1 = 3 and x2 = 7 and for all other k we have that xk = 5xk−1 − 6xk−2. Using strong induction prove that for all n that xn = 2n + 3n−1

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