Principle of Mathematical Induction
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If it is known that (1) some statement is true for n = 1 (2) assumption that statement is true for n implies that the statement is true for (n + 1) then the statement is true for all positive integers Modifications of the Principle of Mathematical Induction • If it is known that (1) some statement is true for n = n0 (positive integer) (2) assumption that statement is true for n implies that the statement is true for (n + 1) then the statement is true for all positive integers greater or equal to n0 • If it is known that (1) some statement is true for n = 1 (2) assumption that statement is true for all positive integers k, 1 k n implies that the statement is true for (n + 1) then the statement is true for all positive integers • (Backward induction) If it is known that (1) some statement is true for n = 1 (2) assumption that statement is true for n > 1 implies that the statement is true for 2n and (n − 1) then the statement is true for all positive integers
Mathematical Induction in Algebra 1. Prove that any positive integer n > 1 is either a prime or can be represented as product of primes factors. 2. Set S contains all positive integers from 1 to 2n. Prove that among any n + 1 numbers chosen from S there are two numbers such that one is a factor of the other. 3. Prove that if (x+1/x) is integer then (xn+1/xn) is also integer for any positive integer n. 4. For sequence of Fibonacci numbers u1 = 1, u2 = 1, uk+1 = uk+uk−1, k = 2, 3, . . . prove the formula uk+m = uk−1um + ukum+1 5. Prove the following identities: (a) 12 + 22 + 32 + · · · + n2 = n(n + 1)(2n + 1)/6 (b) 13 + 23 + 33 + · · · + n3 = n2(n + 1)2/4 (c) 1 × 2 × 3 + 2 × 3 × 4 + · · · + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3)/4 (d) 1 × 1! + 2 × 2! + · · · + n × n! = (n + 1)! − 1 6. Prove the following divisibilities: (a) 6/n3 + 5n (b) 7/(62n−1 + 1) (c) 3n+1/23n + 1 7. Prove the following inequalities: (a) 1/(n + 1) + 1/(n + 2) + 1/(n + 3) + · · · + 1/2n > 13/24 (n > 1) (b) 1/12 + 1/22 + 1/32 + · · · + 1/n2 < id="dwlinks" rel="nofollow" href="http://www.math.toronto.edu/oz/turgor/Induction.pdf" target="_blank">here
Mathematical Induction in Algebra 1. Prove that any positive integer n > 1 is either a prime or can be represented as product of primes factors. 2. Set S contains all positive integers from 1 to 2n. Prove that among any n + 1 numbers chosen from S there are two numbers such that one is a factor of the other. 3. Prove that if (x+1/x) is integer then (xn+1/xn) is also integer for any positive integer n. 4. For sequence of Fibonacci numbers u1 = 1, u2 = 1, uk+1 = uk+uk−1, k = 2, 3, . . . prove the formula uk+m = uk−1um + ukum+1 5. Prove the following identities: (a) 12 + 22 + 32 + · · · + n2 = n(n + 1)(2n + 1)/6 (b) 13 + 23 + 33 + · · · + n3 = n2(n + 1)2/4 (c) 1 × 2 × 3 + 2 × 3 × 4 + · · · + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3)/4 (d) 1 × 1! + 2 × 2! + · · · + n × n! = (n + 1)! − 1 6. Prove the following divisibilities: (a) 6/n3 + 5n (b) 7/(62n−1 + 1) (c) 3n+1/23n + 1 7. Prove the following inequalities: (a) 1/(n + 1) + 1/(n + 2) + 1/(n + 3) + · · · + 1/2n > 13/24 (n > 1) (b) 1/12 + 1/22 + 1/32 + · · · + 1/n2 < id="dwlinks" rel="nofollow" href="http://www.math.toronto.edu/oz/turgor/Induction.pdf" target="_blank">here
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