Textbook Question
Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a(sub n) to find a(sub 8), the eighth term of the sequence. 1, 2, 4, 8, ...
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 39
Find 1+2+3+4+...+ 100, the sum of the first 100 natural numbers.
Verified step by step guidance1
Recognize that the problem asks for the sum of the first 100 natural numbers, which is an arithmetic series where the first term \(a_1 = 1\) and the last term \(a_n = 100\).
Recall the formula for the sum of the first \(n\) natural numbers: \(S_n = \frac{n(n+1)}{2}\), where \(n\) is the number of terms.
Identify that in this problem, \(n = 100\) because we are summing from 1 to 100.
Substitute \(n = 100\) into the formula to set up the expression: \(S_{100} = \frac{100(100+1)}{2}\).
Simplify the expression step-by-step to find the sum, but do not calculate the final numeric value as per instructions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. In this problem, the numbers 1, 2, 3, ..., 100 form an arithmetic sequence with a common difference of 1.
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Formula for the Sum of the First n Natural Numbers
The sum of the first n natural numbers can be found using the formula S = n(n + 1)/2. This formula provides a quick way to calculate the sum without adding each number individually.
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Natural Numbers
Natural numbers are the set of positive integers starting from 1 (1, 2, 3, ...). Understanding that the problem involves the first 100 natural numbers helps identify the sequence and apply the appropriate summation formula.
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