/*! 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 64 Find the indicated quantities fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 19: Problem 64

Find the indicated quantities for the appropriate arithmetic sequence.Show that the sum of the first \(n\) positive odd integers is \(n^{2}\).

Short Answer

Expert verified
The sum of the first \(n\) positive odd integers is \(n^2\).

Step by step solution

01

Understand the Sequence

The sequence of the first positive odd integers is 1, 3, 5, 7,... These numbers form an arithmetic sequence where the first term \(a_1\) is 1, and the common difference \(d\) is 2.
02

Use the Sum Formula for Arithmetic Sequences

The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by the formula \(S_n = \frac{n}{2} (2a_1 + (n-1)d)\). Here, \(a_1 = 1\) and \(d = 2\).
03

Substitute the Known Values

Substitute \(a_1 = 1\) and \(d = 2\) into the formula: \[S_n = \frac{n}{2} (2 \times 1 + (n-1) \times 2) = \frac{n}{2} (2 + 2n - 2) = \frac{n}{2} (2n)\].
04

Simplify the Expression

Simplify the expression \(\frac{n}{2} (2n)\) to get \(n^2\). This shows that the sum of the first \(n\) positive odd integers is indeed \(n^2\).
05

Verify with an Example

Let's verify with \(n=3\). The first three positive odd integers are 1, 3, and 5. Their sum is \(1 + 3 + 5 = 9\). Since \(n = 3\), \(n^2 = 3^2 = 9\), which matches the sum calculated.

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.

Sum of Odd Integers
When we talk about the sum of odd integers, we are focusing on a special subset of numbers. These numbers do not have any fractions or even parts, and include numbers like 1, 3, 5, 7, and so on.
The addition of these integers, especially when they are consecutive odd numbers starting from 1, forms a unique pattern that can actually be predicted using a mathematical formula.
One fascinating result is that the sum of the first "n" positive odd integers equals \(n^2\).
For example, if you take the first four odd integers (1, 3, 5, and 7), their sum is 16. Coincidentally, 16 is exactly \(4^2\), because "n" in this case is 4.
Sum Formula for Arithmetic Sequence
Arithmetic sequences have a specific form and structure. They have a starting number, known as the first term, and a consistent step between terms, known as the common difference.
The formula for finding the sum \(S_n\) of the first "n" terms in an arithmetic sequence is:
\[S_n = \frac{n}{2} (2a_1 + (n-1)d)\]
Where:
  • \(n\) is the number of terms,
  • \(a_1\) is the first term,
  • \(d\) is the common difference.
By plugging in the correct values for any given sequence, you can use this formula to find the sum of its terms efficiently.
Positive Odd Integers
Positive odd integers are part of the whole number system that start at one and increase by skipping every other number.
These numbers never end in zero or another even number. Instead, they always increase by increments of two.
Some specific positive odd integers include:
  • 1
  • 3
  • 5
  • 7
  • 9
These form an arithmetic sequence where the first term is 1 and each subsequent term increases by the common difference. They鈥檙e fundamental in demonstrating important mathematical principles, particularly in forming patterns and sequences.
Common Difference in Arithmetic Sequences
The term "common difference" is crucial in understanding arithmetic sequences.
It refers to the regular interval at which numbers increase as you move from one term of the sequence to the next.
For example, in the sequence of positive odd integers (1, 3, 5, 7, 9...), the common difference "d" is 2.
This means to get from any odd integer in the sequence to the next, you need to add 2.
Identifying the common difference helps in predicting future terms of the sequence, and it plays an essential role when calculating the sum of the sequence using the sum formula.

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

Find the indicated quantities.Measurements show that the temperature of a distant star is presently \(9800^{\circ} \mathrm{C}\) and is decreasing by \(10 \%\) every 800 years. What will its temperature be in 4000 years?

A helium-filled balloon rose \(120 \mathrm{ft}\) in 1.0 min. Each minute after that, it rose \(75 \%\) as much as in the previous minute. What was its maximum height?

Solve the given problems by use of the sum of an infinite geometric series. If a major league baseball team can make it to the World Series, it can be a great financial boost to the economy of their city. Let us assume, if their team plays in the World Series, that tourists will spend 20,000,000 dollar for hotels, restaurants, local transporation, tickets for the Series and city attractions, and so forth. We now assume that \(75 \%\) of this money will be spent in a second round of spending in the city by those who received it. A third round, fourth round, fifth round, and so on of \(75 \%\) spending will follow. Now assuming this continues indefinitely, what is the total amount of this spending, which in effect is added to the economy of the city because of the World Series? This problem illustrates one of the major reasons a city wants to host major events that attract many tourists.

Solve the given problems. In the expansion of \((a-x)^{n},\) where \(n\) is a positive integer, show that the sum of the coefficients is zero.

If a major league baseball team can make it to the World Series, it can be a great financial boost to the economy of their city. Let us assume, if their team plays in the World Series, that tourists will spend \(\$ 20,000,000\) for hotels, restaurants, local transporation, tickets for the Series and city attractions, and so forth. We now assume that \(75 \%\) of this money will be spent in a second round of spending in the city by those who received it. A third round, fourth round, fifth round, and so on of \(75 \%\) spending will follow. Now assuming this continues indefinitely, what is the total amount of this spending, which in effect is added to the economy of the city because of the World Series? This problem illustrates one of the major reasons a city wants to host major events that attract many tourists.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.