Textbook Question
Evaluate each expression.
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 25
Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
Verified step by step guidance1
Identify the first term \(a_1\) and the common difference \(d\) of the arithmetic sequence. These are essential to use the sum formula.
Recall the formula for the sum of the first \(n\) terms of an arithmetic sequence:
\[ S_n = \frac{n}{2} \times (2a_1 + (n - 1)d) \]
Determine the number of terms \(n\) you need to sum, based on the problem's given range or limits.
Substitute the values of \(a_1\), \(d\), and \(n\) into the sum formula to set up the expression for \(S_n\).
Simplify the expression step-by-step to find the sum of the first \(n\) terms, but do not calculate the final numeric value unless asked.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Sequence
An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. Understanding this pattern is essential for identifying terms and applying formulas related to sums.
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Arithmetic Sequences - General Formula
Sum of the First n Terms of an Arithmetic Sequence
This formula calculates the total of the first n terms in an arithmetic sequence. It is given by S_n = n/2 * (2a_1 + (n-1)d), where a_1 is the first term and d is the common difference, enabling efficient summation without adding each term individually.
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Summation Notation (Sigma Notation)
Summation notation uses the Greek letter sigma (∑) to represent the sum of a sequence of terms. It provides a concise way to express adding multiple terms, which is crucial for interpreting and solving problems involving series.
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Related Practice
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Evaluate each factorial expression. 16!/2!14!