Domanda del libro di testo
Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 14 terms of the geometric sequence: - 3/2, 3, - 6, 12, ...
909
views
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 28
In Exercises 23–28, evaluate each factorial expression. (2n+1)!/(2n)!
Guida verificata passo dopo passo1
Identify the factorial expressions in the problem: \((2n+1)!\) and \((2n)!\).
Recall the definition of a factorial: \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).
Express \((2n+1)!\) in terms of \((2n)!\): \((2n+1)! = (2n+1) \times (2n)!\).
Substitute the expression for \((2n+1)!\) into the original problem: \(\frac{(2n+1) \times (2n)!}{(2n)!}\).
Simplify the expression by canceling out \((2n)!\) from the numerator and the denominator, leaving \(2n+1\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Factorial Definition
A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations, combinations, and various mathematical calculations, making them fundamental in algebra and combinatorics.
Video consigliato:
5:22
Factorials
Properties of Factorials
Factorials have specific properties that simplify calculations, such as n! = n × (n-1)! and 0! = 1. These properties allow for the reduction of factorial expressions, which is essential when evaluating complex factorial ratios like (2n+1)!/(2n)!. Understanding these properties helps in manipulating and simplifying expressions effectively.
Video consigliato:
5:22Factorials
Ratio of Factorials
The ratio of two factorials, such as (2n+1)!/(2n)!, can be simplified by canceling common terms. This involves recognizing that (2n+1)! = (2n+1) × (2n)!, allowing for the expression to reduce to (2n+1). This concept is crucial for evaluating factorial expressions efficiently and accurately.
Video consigliato:
5:22Factorials
Pratica correlata
Domanda del libro di testo
Use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of n(n + 1)(n + 2).
691
views
Domanda del libro di testo
Use mathematical induction to prove that each statement is true for every positive integer n.
736
views
Domanda del libro di testo
Use the Fundamental Counting Principle to solve Exercises 29–40. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or sport). In how many ways can you order the car?
720
views
Domanda del libro di testo
Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises 25–30. Find the sum of the first 11 terms of the geometric sequence: 3, - 6, 12, - 24, ...
945
views
Domanda del libro di testo
Evaluate each expression.
564
views