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.

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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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