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 12 terms of the geometric sequence: 2, 6, 18, 54, ...
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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 25
Evaluate each expression.
Guida verificata passo dopo passo1
Identify the combinations in the expression: \$7C3$ and \$5C4$. Recall that the combination formula is given by \(nCr = \frac{n!}{r!(n-r)!}\).
Write out each combination using the formula: \(7C3 = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!}\) and \(5C4 = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!}\).
Calculate each combination separately by simplifying the factorial expressions. For example, simplify \$7!\(, \)3!$, and \$4!$ as needed to find \$7C3$, and similarly for \$5C4$.
Evaluate the factorial fraction \(\frac{98!}{96!}\) by simplifying it. Remember that \(\frac{98!}{96!} = 98 \times 97\) because the factorial terms cancel out except for the last two factors.
Substitute the values of \$7C3\(, \)5C4$, and \(\frac{98!}{96!}\) back into the original expression and perform the subtraction to complete the evaluation.
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Combination Formula
A combination represents the number of ways to choose a subset of items from a larger set without regard to order. It is calculated using the formula nCr = n! / [r!(n-r)!], where n is the total number of items and r is the number chosen.
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5:22
Combinations
Factorials
A factorial, denoted by n!, is the product of all positive integers from 1 up to n. Factorials are fundamental in permutations and combinations, simplifying expressions involving counting and arrangements.
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5:22Factorials
Simplifying Factorial Expressions
When evaluating expressions involving factorials, it is often helpful to cancel common terms in the numerator and denominator. This simplification reduces computational complexity and helps in accurately calculating combinations or other factorial-based expressions.
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5:22Factorials
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