Domanda del libro di testo
Write the first three terms in each binomial expansion, expressing the result in simplified form. (x - 2y)10
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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 33
Find each indicated sum.
Guida verificata passo dopo passo1
Identify the sum notation: you need to find the sum of the expression \(k(k+4)\) as \(k\) goes from 1 to 5, which is written as \(\sum_{k=1}^{5} k(k+4)\).
Expand the expression inside the summation: \(k(k+4) = k^2 + 4k\).
Rewrite the sum as the sum of two separate sums: \(\sum_{k=1}^{5} (k^2 + 4k) = \sum_{k=1}^{5} k^2 + \sum_{k=1}^{5} 4k\).
Use the properties of summation to factor out constants: \(\sum_{k=1}^{5} 4k = 4 \sum_{k=1}^{5} k\).
Calculate each sum separately using formulas: \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\) and \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\), then substitute \(n=5\) and add the results.
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Summation Notation (Sigma Notation)
Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms. The expression 5Σk=1 means to sum the terms as k goes from 1 to 5. Understanding this notation is essential to correctly evaluate the sum.
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Interval Notation
Evaluating Polynomial Expressions
Each term in the sum involves evaluating the polynomial k(k+4) for each integer k from 1 to 5. This requires substituting values of k into the expression and simplifying before summing the results.
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03:11Evaluating Algebraic Expressions
Properties of Finite Sums
Finite sums can be broken down into sums of simpler expressions, such as sums of k and sums of constants. Using properties like linearity of summation helps simplify calculations by separating and summing individual parts.
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5:36Change of Base Property
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