Textbook Question
Find the term indicated in each expansion. (x2 + y3)8; sixth term
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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 45
Write out the first three terms and the last term. Then 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 arithmetic sequence given by the general term \(a_i = 5i + 3\), where \(i\) ranges from 1 to 17.
Find the first three terms by substituting \(i = 1, 2, 3\) into the formula:
\(a_1 = 5(1) + 3\),
\(a_2 = 5(2) + 3\),
\(a_3 = 5(3) + 3\).
Find the last term by substituting \(i = 17\) into the formula:
\(a_{17} = 5(17) + 3\).
Recall the formula for the sum of the first \(n\) terms of an arithmetic sequence:
\(S_n = \frac{n}{2} (a_1 + a_n)\),
where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.
Substitute \(n = 17\), the first term \(a_1\), and the last term \(a_{17}\) into the sum formula to express the sum \(S_{17}\) without calculating the final value.
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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. For example, in the sequence defined by 5i + 3, the difference between consecutive terms is constant (5). Understanding this helps identify the terms and their pattern.
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Arithmetic Sequences - General Formula
Sum of the First n Terms of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence can be found using the formula S_n = n/2 (a_1 + a_n), where a_1 is the first term and a_n is the nth term. This formula simplifies the process of adding many terms without listing them all.
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Evaluating Summation Notation
Summation notation (Σ) represents the sum of terms defined by a formula over a range of indices. Here, it sums the expression 5i + 3 from i = 1 to 17. Understanding how to interpret and expand this notation is essential for finding specific terms and the total sum.
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Related Practice
Textbook Question
Find the sum of the even integers between 21 and 45.
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Textbook Question
In Exercises 45-46, it is equally probable that the pointer on the spinner shown will land on any one of the eight regions, numbered 1 through 8. If the pointer lands on a borderline, spin again. Find the probability that the pointer will stop on an odd number or a number less than 6.
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Textbook Question
Find the term indicated in each expansion. (x − 1/2)9; fourth term
Textbook Question
Express each sum using summation notation. Use as the lower limit of summation and for the index of summation.
Textbook Question
Find the sum of the odd integers between 30 and 54.