/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 REASONING Find the sum of the po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 61

REASONING Find the sum of the positive odd integers less than 300 . Explain your reasoning.

Short Answer

Expert verified
The sum of the positive odd integers less than 300 is 22500.

Step by step solution

01

Identify the sequence

Firstly, the sequence needs to be identified. We're dealing with odd numbers, which could be represented as 1, 3, 5, 7, etc.
02

Find the common difference

In this arithmetic sequence, the common difference\(d\) is 2, as each succeeding term is 2 more than the preceding term.
03

Calculate the number of terms

The last term\(l\) in the sequence less than 300 is 299, and the first term\(a\) is 1. The formula to calculate the number of terms \(n\) in an arithmetic series is \((l - a) / d + 1\). So plug the values into equation, \((299 - 1) / 2 + 1 = 150\). Hence, there are 150 terms.
04

Calculate the sum

To find the sum \(S\) of an arithmetic series, the formula \(n/2 * (a + l)\) is used. Here, \(n = 150\), \(a = 1\) and \(l = 299\). Substitute these into the formula, we get \(150/2 * (1 + 299) = 22500\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Odd Numbers
Odd numbers are integers that are not divisible by 2. This means when you divide an odd number by 2, you'll get a remainder of 1. Examples include 1, 3, 5, 7, 9, and so on.
Unlike even numbers, which alternate with odd numbers on the number line, odd numbers are critical in various mathematical patterns and sequences.
They are regularly spaced by a difference of 2, which makes them an interesting subject for arithmetic sequences.
Arithmetic Sequence
An arithmetic sequence is a list of numbers where each term after the first is the sum of the previous term and a constant, called the "common difference."
For example, in the sequence of odd numbers 1, 3, 5, 7, each number is obtained by adding 2 to the previous number.
This helps determine not just the possible numbers in the sequence, but also how many such numbers can be included in a given range.
Common Difference
In an arithmetic sequence, the common difference is the consistent amount that separates each pair of successive terms.
For the odd numbers sequence, the common difference is 2 since each odd number is two more than the last.
Understanding the common difference allows us to predict future numbers in the sequence and calculate various properties, such as the total number of terms or their sum.
Sequence Reasoning
Sequence reasoning involves understanding the pattern and rules that define a sequence, which allows us to make predictions or solve problems involving sequences.
For instance, by recognizing the pattern in the odd numbers up to 299, we can use the formula for the sum of an arithmetic series to find the total of these numbers.
In this exercise, knowing that the series starts at 1 and ends at 299, with a common difference of 2, allows us to calculate quickly and accurately using established mathematical techniques.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The rule for a recursive sequence is as follows. $$ \begin{aligned} &f(1)=3, f(2)=10 \\ &f(n)=4+2 f(n-1)-f(n-2) \end{aligned} $$ a. Write the first five terms of the sequence. b. Use finite differences to find a pattern. What type of relationship do the terms of the sequence show? c. Write an explicit rule for the sequence.

You want to save \(\$ 500\) for a school trip. You begin by saving a penny on the first day. You save an additional penny each day after that. For example, you will save two pennies on the second day, three pennies on the third day, and so on. a. How much money will you have saved after 100 days? b. Use a series to determine how many days it takes you to save \(\$ 500\).

You borrow \(2000 at 9% annual interest compounded monthly for 2 years. The monthly payment is \)91.37. a. Find the balance after the fifth payment. b. Find the amount of the last payment.

Write the repeating decimal as a fraction in simplest form. \(0.222 \ldots\)

Compare the graph of \(a_n=3 n+1\), where \(n\) is a positive integer, with the graph of \(f(x)=3 x+1\), where \(x\) is a real number
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.