Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 33

Capitolo 9, Problema 33

Use mathematical induction to prove that each statement is true for every positive integer n. (ab)ⁿ = aⁿ bⁿ

Guida verificata passo dopo passo

Step 1: **State the proposition** you want to prove using mathematical induction. Let $ P(n) $ be the statement $ (ab)^n = a^n b^n $ for a positive integer $ n $.

Step 2: **Base case:** Verify that $ P(1) $ is true. Substitute $ n = 1 $ into the statement to check if $ (ab)^1 = a^1 b^1 $ holds.

Step 3: **Inductive hypothesis:** Assume that $ P(k) $ is true for some positive integer $ k $, i.e., assume $ (ab)^k = a^k b^k $.

Step 4: **Inductive step:** Use the inductive hypothesis to prove that $ P(k+1) $ is true. Start with $ (ab)^{k+1} = (ab)^k (ab) $ and then substitute the inductive hypothesis to rewrite it as $ a^k b^k (ab) $.

Step 5: **Simplify the expression** from Step 4 by using the associative and commutative properties of multiplication to show that $ a^k b^k (ab) = a^{k+1} b^{k+1} $, thus proving $ P(k+1) $. Conclude that by mathematical induction, the statement is true for all positive integers $ n $.

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Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.

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Properties of Exponents

The properties of exponents describe how to manipulate expressions involving powers. One key property is that (ab)^n = a^n * b^n, meaning the nth power of a product equals the product of the nth powers. Understanding this property is essential to prove the given statement using induction.

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Rational Exponents

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In this proof, it is necessary to rewrite (ab)^(k+1) as (ab)^k * (ab) and then apply the induction hypothesis and exponent rules to show equality. Mastery of these skills helps in constructing clear and logical proofs.

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