Domanda del libro di testo
Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a60 when a1 = 35, d = -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 21
Use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3
Guida verificata passo dopo passo1
Step 1: Define the statement to prove by induction. Let \( P(n) \) be the statement \( 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + n(n + 1) = \frac{n(n + 1)(n + 2)}{3} \). We want to prove \( P(n) \) is true for all positive integers \( n \).
Step 2: Verify the base case \( n = 1 \). Substitute \( n = 1 \) into both sides of the equation and check if the left-hand side equals the right-hand side.
Step 3: Assume the induction hypothesis \( P(k) \) is true for some positive integer \( k \), that is, assume \( 1 \cdot 2 + 2 \cdot 3 + \cdots + k(k + 1) = \frac{k(k + 1)(k + 2)}{3} \).
Step 4: Prove the statement \( P(k + 1) \) is true using the induction hypothesis. Start with the left-hand side of \( P(k + 1) \): \( 1 \cdot 2 + 2 \cdot 3 + \cdots + k(k + 1) + (k + 1)(k + 2) \). Substitute the induction hypothesis for the sum up to \( k(k + 1) \).
Step 5: Simplify the expression obtained in Step 4 and show that it equals the right-hand side of \( P(k + 1) \), which is \( \frac{(k + 1)(k + 2)(k + 3)}{3} \). This completes the induction step and proves the statement 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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Types of Slope
Summation of Sequences
Summation of sequences involves finding a formula that represents the sum of terms in a sequence. In this problem, the sum is of products of consecutive integers, and the goal is to express this sum in a closed form, which simplifies calculations and helps in proving the formula by induction.
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Algebraic Manipulation
Algebraic manipulation refers to the process of simplifying, expanding, or factoring expressions to transform them into a desired form. In induction proofs, it is essential to manipulate expressions correctly to show that the formula for n+1 follows logically from the formula for n.
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05:09Introduction to Algebraic Expressions
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