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
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x − 1)5
Guida verificata passo dopo passo1
Recall the Binomial Theorem formula: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \), where \(\binom{n}{k}\) is the binomial coefficient.
Identify the terms in the binomial: here, \(a = x\), \(b = -1\), and \(n = 5\).
Write the expansion using the formula: \( (x - 1)^5 = \sum_{k=0}^5 \binom{5}{k} x^{5-k} (-1)^k \).
Calculate each term by finding \(\binom{5}{k}\), raising \(x\) to the power \(5-k\), and \((-1)\) to the power \(k\), then multiply these values together for each \(k\) from 0 to 5.
Combine all the terms from \(k=0\) to \(k=5\) to write the full expanded expression, and simplify the signs and coefficients where possible.
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Binomial Theorem
The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n equals the sum of terms C(n, k) * a^(n-k) * b^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the binomial repeatedly.
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Special Products - Cube Formulas
Binomial Coefficients
Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term in the expansion.
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03:41Special Products - Cube Formulas
Simplifying Polynomial Expressions
After expanding a binomial using the Binomial Theorem, simplifying involves combining like terms and reducing powers where possible. This step ensures the final expression is in its simplest form, making it easier to interpret or use in further calculations. Simplification also includes applying negative signs correctly when terms involve subtraction.
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05:07Simplifying Algebraic Expressions
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