Problem 28 In Exercises $11-30,$ use math... [FREE SOLUTION]

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College Algebra

Robert Blitzer

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 8: Problem 28

In Exercises $11-30,$ use mathematical induction to prove that each statement is true for every positive integer $n$ 3 is a factor of $n(n+1)(n-1)$

Short Answer

Expert verified

By mathematical induction, it is proved that for every positive integer $ n $, 3 is a factor of the product $ n(n+1)(n-1) $.

Step by step solution

Base case for n=1

First, check if the statement holds true for the base case of $ n = 1 $. The expression becomes $1*(1+1)*(1-1) = 2*0 = 0$. And indeed, 3 is a factor of 0.

Assume true for n=k

Suppose the statement is true for $ n = k $. This implies that 3 is a factor of $ k(k+1)(k-1) $. Let $ k(k+1)(k-1) = 3p $ for some positive integer $ p $. That is, $ k(k+1)(k-1) $ is divisible by 3.

Show true for n=k+1

To complete the proof, show that the statement remains true for $ n = k+1 $. The expression becomes: $(k+1)((k+1)+1)((k+1)-1) = (k+1)(k+2)(k)$. To see if this is divisible by 3, express it in terms of $ k(k+1)(k-1) $. This can be written as $ k(k+1)(k-1) + 3k(k+1) = 3p + 3k(k+1)$, which is divisible by 3, because both terms are divisible by 3.

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91影视

In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) 3 is a factor of \(n(n+1)(n-1)\)

Short Answer

Step by step solution

Base case for n=1

Assume true for n=k

Show true for n=k+1

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