Comprehensive Guide to Proving Products Using Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the validity of statements involving products. This guide provides detailed steps and examples to help you understand and apply mathematical induction with products.

 

Statement to Prove:

i=1n (1 – 1/(i+1)) = 1/(n+1)

Proof Using Mathematical Induction

Base Case:

For n = 1:

i=11 (1 – 1/(i+1)) = 1 – 1/(1+1) = 1 – 1/2 = 1/2

The right-hand side is:

1/(1+1) = 1/2

Thus, the base case holds for n = 1.

Inductive Hypothesis:

Assume the statement is true for n = k:

i=1k (1 – 1/(i+1)) = 1/(k+1)

Inductive Step:

We need to prove the statement holds for n = k+1:

i=1k+1 (1 – 1/(i+1)) = 1/(k+2)

Using the inductive hypothesis:

i=1k+1 (1 – 1/(i+1)) = (∏i=1k (1 – 1/(i+1))) (1 – 1/(k+2))

By the inductive hypothesis, the first product is:

1/(k+1)

Substituting this into the equation:

1/(k+1) * (1 – 1/(k+2)) = 1/(k+1) * (k+1)/(k+2) = 1/(k+2)

Thus, the statement holds for n = k+1.

Conclusion:

Since the base case holds for n = 1, and the inductive step proves that if the statement is true for n = k, it is also true for n = k+1, by the principle of mathematical induction, the statement is true for all n ≥ 1.

 

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