Textbook Question
Find each indicated sum.
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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 37
Write the first three terms in each binomial expansion, expressing the result in simplified form. (y3 − 1)20
Verified step by step guidance1
Identify the binomial expression and the exponent: the expression is \((y^{3} - 1)^{20}\), where \(a = y^{3}\), \(b = -1\), and \(n = 20\).
Recall the Binomial Theorem formula for expansion: \(\displaystyle (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
Write the first three terms of the expansion by substituting \(k = 0, 1, 2\) into the formula:
\[\text{Term}_k = \binom{20}{k} (y^{3})^{20-k} (-1)^k\]
Simplify each term by calculating the binomial coefficients, powers of \(y^{3}\) (which become \(y^{3(20-k)}\)), and powers of \(-1\), then express each term in simplified form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Theorem
The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum involving terms with binomial coefficients. Each term is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient representing combinations. This theorem is essential for finding specific terms in the expansion without fully multiplying the expression.
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Binomial Coefficients
Binomial coefficients, denoted as C(n, k) or "n choose k," count 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. Understanding these coefficients is crucial for determining the coefficients of each term in the expansion.
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Simplifying Powers and Expressions
After applying the Binomial Theorem, each term often involves powers of variables and constants. Simplifying these powers, such as (y^3) raised to a power, requires using exponent rules like (a^m)^n = a^(m*n). Simplification ensures the final terms are expressed in their simplest form, making the expansion clear and concise.
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Related Practice
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Find the sum of the first 50 terms of the arithmetic sequence: -10, -6, -2, 2,....
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Find the sum of each infinite geometric series. 1 + 1/3 + 1/9 + 1/27 + ...
Textbook Question
Find each indicated sum.
Textbook Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a(sub 6) when a(sub 1) = 16, r = 1/2
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Textbook Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence. Find a5 when a1 = -3, r = 2