Textbook Question
In Exercises 23–28, evaluate each factorial expression. (2n+1)!/(2n)!
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 29
Find each indicated sum.
Verified step by step guidance1
Identify the summation notation given: \(\sum_{i=1}^{6} 5i\). This means you need to find the sum of the expression \$5i\( as \)i$ goes from 1 to 6.
Rewrite the summation by expanding the terms: \(5(1) + 5(2) + 5(3) + 5(4) + 5(5) + 5(6)\).
Factor out the constant 5 from the summation: \(5 \sum_{i=1}^{6} i\).
Use the formula for the sum of the first \(n\) natural numbers: \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\). Here, \(n=6\).
Substitute \(n=6\) into the formula and multiply by 5: \(5 \times \frac{6 \times (6+1)}{2}\). This expression represents the sum you need to calculate.
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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. It specifies the index of summation, the lower and upper limits, and the expression to be summed. For example, Σ from i=1 to 6 of 5i means adding 5 times each integer from 1 to 6.
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Interval Notation
Arithmetic Series
An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, 5i forms an arithmetic sequence with a common difference of 5. Understanding this helps in applying formulas to find the sum efficiently.
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Formula for the Sum of an Arithmetic Series
The sum of the first n terms of an arithmetic series can be found using the formula S_n = n/2 (first term + last term). This formula simplifies the calculation by avoiding term-by-term addition, making it useful for quickly finding sums like Σ 5i from i=1 to 6.
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Related Practice
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