MTH 3318 Solutions to Induction Problems Fall 2009
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1 Pat Rossi Instructions. MTH 338 Solutions to Induction Problems Fall 009 Name Prove the following by mathematical induction. Set (): n n(n+) i n(n+) Step #: Show that the proposition is true for n. ()(()+) i True. Step #: Assume that the proposition is true for n k, and prove that the proposition is true for n k +. i k(k+) is true for some natural number k, and prove that i (k+)((k+)+) is true. (Equivalently, prove that i k +3k+.) P k+ i k (k +) i +(k +) +(k +) k +k + k+ k +3k+. i.e., i k +3k+. i n(n+) for all natural numbers, n. k(k+) + (k+) Set (): Given that x + x x + x (the Triangle Inequality); Prove by induction that: x + x + x x n x + x + x x n (the General Triangle Inequality). Step #: Show that the proposition is true for n. x x. True. Step #: Assume that the proposition is true for n k, and prove that the proposition is true for n k +. i.e., Assume that x + x + x x k x + x + x x k and show that x + x + x x k + x k+ x + x + x x k + x k+.
2 (x + x + x x k )+x k+ x + x + x x k + x k+ from given x + x + x x k + x k+. by Ind. Hyp. i.e., x + x + x x k + x k+ x + x + x x k + x k+. Hence, x + x + x x n x + x + x x n for all natural numbers, n. Set (5): ( + x) n +nx for any natural number n and any real number x. ( + x) +x +()x. Assume true for n k, and show true for n k + i.e., Assume that ( + x) k +kx for some natural number k, and show that ( + x) k+ +(k +)x ( + x) k+ (+x) k ( + x) ( + kx)(+x) +(k +)x + kx {z} kx 0 i.e., ( + x) k+ +(k +)x +(k +)x +kx + x + kx Hence, ( + x) n +nx for all natural numbers n and any real number x Remark Our proof hinged on two subtle points: First, since k is a natural number (hence greater than zero) and x 0 for ALL real numbers x, it follows that kx 0. Second, since it is given that x (or equivalently, ( + x) 0), the direction of the inequality, ( + x) k +kx, is preserved when both sides are multiplied by ( + x) during the application of the induction hypothesis.
3 Set (6): For 0 a b; prove that a n b n.. a a b given i.e., a b b. Assume true for n k, and show true for n k + i.e., Assume that a k b k for some natural number k, and show that a k+ b k+ a k+ a k a i.e., a k+ b k+ b k a by Ind. Hyp. b k b {z} a b Hence, a n b n for all natural numbers, n. b k+ Set (7): (n ) n (i ) n (i ) ( () )(). Assume true for n k, and show true for n k + (i ) k for some natural number k, and show that (i ) (k +) (i ) + ( (k +) ) k +((k +) ) (i ) k +k +(k +) i.e., (i ) (k +) (i ) n for all natural numbers, n. 3
4 Set (): n 3 n (n+) i3 n (n+) i3 () 3 () (()+). Assume true for n k, and show true for n k + i3 k (k+) for some natural number k, and show that i3 (k+) ((k+)+) i.e., i3 (k+) (k+) i3 i 3 +(k +) 3 k (k +) +(k +) 3 (k+) [k +(k +)] (k+) [k +k +] (k+) (k+) i.e., P n i3 n (n+) for all natural numbers, n. Set (7): (n )(n+) j (j )(j+) n n+ j n n+ (() )(()+) ()(3) 3 ()+. Assume true for n k, and show true for n k + j show that j i.e., j (j )(j+) k+ (k+)+ k+ (j )(j+) k+3 k (j )(j+) k+ k (k+) + (k+)3 for some natural number k, and
5 j i.e., j j P k (j )(j+) j k+ (j )(j+) k+3 (j )(j+) + (j )(j+) ((k+) )((k+)+) k k+ + (k+)(k+3) k k+ k+3 k+3 + (k+)(k+3) (k+)(k+) (k+)(k+3) (k+) (k+3) n n+ () k +3k+ (k+)(k+3) for all natural numbers, n. Set (): (n ) 3 < n < n 3 for all natural numbers, n. (i )3 < n < P n i3 (i )3 (() ) 3 0< < 3 i3. Assume true for n k, and show true for n k + (i )3 < k < P k i3 for some natural number k, and show that (i )3 < (k+) < i3 (i )3 (i ) 3 + k 3 < k + k3 k + k3 k +k 3 < k +k 3 +6k +k+ (k+) i.e., (i )3 < (k+) Next i3 i 3 +(k +) 3 > k +(k +)3 k + (k+)3 k +k 3 +k +k+ > k +k 3 +6k +k+ (k+) i.e., i3 > (k+) 5
6 (i )3 < n < P n i3 for all natural numbers, n. 6
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