Textbook Question
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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Table of contents
- 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
Chapter 9, Problem 33
Find each indicated sum.
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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Related Practice
Textbook Question
Use the Fundamental Counting Principle to solve Exercises 29–40. An ice cream store sells two drinks (sodas or milk shakes) in four sizes (small, medium, large, or jumbo) and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?
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Textbook Question
Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. an= an-1 -10, a1 = 30
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Textbook Question
In Exercises 31–34, write the first five terms of each geometric sequence. a1 = 1/2, r = 1/2
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Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. (ab)n = an bn
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Textbook Question
Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.
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