a) For every positive integer n, n! is defined to be n! = 1 · 2 · 3 · … · n. Use mathematical induction to show that 1 ·1! + 2 ·2! + 3 ·3! + … + n · n! = (n + 1)! − 1 for all positive integers. (Hint: note that (n + 1)! = n!(n + 1).) b) Use mathematical induction to show that the product of three consecutive positive integers is always divisible by 6.
a) For every positive integer n, n! is defined to be n! = 1 · 2 · 3 · … · n. Use mathematical induction to show that 1 ·1! + 2 ·2! + 3 ·3! + … + n · n! = (n + 1)! − 1 for all positive integers. (Hint: note that (n + 1)! = n!(n + 1).) b) Use mathematical induction to show that the product of three consecutive positive integers is always divisible by 6.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 42E
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a) For every positive integer n, n! is defined to be n! = 1 · 2 · 3 · … · n. Use mathematical induction to show that 1 ·1! + 2 ·2! + 3 ·3! + … + n · n! = (n + 1)! − 1 for all positive integers. (Hint: note that (n + 1)! = n!(n + 1).)
b) Use mathematical induction to show that the product of three consecutive positive integers is always divisible by 6.
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