Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 24

Chapter 9, Problem 24

Find 3 + 6 + 9 + ... + 300, the sum of the first 100 positive multiples of 3.

Verified step by step guidance

Recognize that the series 3 + 6 + 9 + ... + 300 is an arithmetic sequence where each term increases by a common difference of 3.

Identify the first term $ a_1 = 3 $ and the last term $ a_n = 300 $.

Determine the number of terms $ n $. Since the problem states the first 100 positive multiples of 3, $ n = 100 $.

Use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} (a_1 + a_n) \], where $ S_n $ is the sum of the first $ n $ terms.

Substitute the known values into the formula: \[ S_{100} = \frac{100}{2} (3 + 300) \] and simplify step-by-step to find the sum.

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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 with a constant difference between consecutive terms. In this problem, the multiples of 3 form an arithmetic sequence starting at 3 with a common difference of 3.

Number of Terms in a Sequence

Determining the number of terms involves identifying how many elements are in the sequence. Here, the problem states the first 100 positive multiples of 3, so the sequence has exactly 100 terms.

Sum of an Arithmetic Series

The sum of an arithmetic series can be found using the formula S = n/2 * (first term + last term), where n is the number of terms. This formula allows efficient calculation of the total sum without adding each term individually.

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