Domanda del libro di testo
Use mathematical induction to prove that the statement is true for every positive integer n. 1 + 4 + 4^2 + ... + 4^(n-1) = ((4^n)-1)/3
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 9, Problema 57
Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2
Guida verificata passo dopo passo1
Step 1: Understand the problem. We are tasked with proving the formula for the sum of the series 5 + 10 + 15 + ... + 5n using mathematical induction. The formula to prove is S(n) = (5n(n+1))/2, where S(n) represents the sum of the first n terms of the series.
Step 2: Base case. Verify the formula for n = 1. Substitute n = 1 into the formula: S(1) = (5 * 1 * (1 + 1)) / 2. Calculate the left-hand side of the series, which is just the first term, 5. Show that both sides are equal, confirming the base case.
Step 3: Inductive hypothesis. Assume the formula is true for some positive integer k, i.e., assume S(k) = 5 + 10 + 15 + ... + 5k = (5k(k+1))/2. This assumption is called the inductive hypothesis and will be used to prove the formula for n = k + 1.
Step 4: Inductive step. Prove the formula for n = k + 1. Add the next term in the series, 5(k+1), to both sides of the inductive hypothesis: S(k+1) = S(k) + 5(k+1). Substitute the assumed formula for S(k) into this equation: S(k+1) = (5k(k+1))/2 + 5(k+1). Simplify the right-hand side to show it matches the formula (5(k+1)((k+1)+1))/2.
Step 5: Conclusion. Since the base case is true and the inductive step has been proven, by mathematical induction, the formula S(n) = (5n(n+1))/2 is true for all positive integers n.
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Mathematical Induction
Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements, typically involving integers. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is essential for proving statements that are asserted for all positive integers.
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Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence, where each term after the first is obtained by adding a constant difference. In the given problem, the series 5 + 10 + 15 + ... + 5n can be expressed as an arithmetic series with a first term of 5, a common difference of 5, and n terms. Understanding the properties of arithmetic series is crucial for deriving the formula for the sum.
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Formula for the Sum of an Arithmetic Series
The formula for the sum of the first n terms of an arithmetic series is given by S_n = n/2 * (a + l), where a is the first term, l is the last term, and n is the number of terms. In this case, the last term can be expressed as 5n, leading to the specific formula for the sum of the series in the problem. Recognizing how to apply this formula is key to proving the statement using induction.
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