/*! 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 39 Find \(1+2+3+4+\dots+100,\) the ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 39

Find \(1+2+3+4+\dots+100,\) the sum of the first 100 natural numbers.

Short Answer

Expert verified
The sum of the first 100 natural numbers is 5050.

Step by step solution

01

Identify the arithmetic series

Firstly, we identify that the sum \(1+2+3+4+\dots+100\) is an arithmetic series. In this case, the first term \(a\) is 1 and the last term \(l\) is 100.
02

Apply the arithmetic series sum formula

The formula for the sum of an arithmetic series is \(S=\frac{n}{2}[a + l]\). In this case, \(n=100\), \(a=1\), and \(l=100\). We substitute these values into the formula, which gives us \[S = \frac{100}{2}[1 + 100].\]
03

Compute the sum

The arithmetic simplifies the formula to \[S = 50 [101] = 5050\]. So, the sum of the first 100 natural numbers is 5050.

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Key Concepts

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

Sum of Natural Numbers
The sum of natural numbers refers to the total you get when you add up a sequence of numbers starting from 1 and continuing consecutively, such as 1, 2, 3, all the way up to a certain number. For example, in the exercise, the task is to find the sum of all numbers from 1 to 100. This sequence is a special type of series known as an arithmetic series. To understand how to find such sums efficiently, we use specific mathematical formulas. This avoids adding individual numbers one by one, which can be tedious for large numbers. Here, we rely on the arithmetic series formula to easily calculate the sum.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant to the previous term. This constant is referred to as the common difference. - The sequence from 1 to 100, which increases by a common difference of 1, is a classic example of an arithmetic sequence.- In this context, the first term (\(a\),) is 1 and the last term (\(l\) ) is 100. - The number of terms (\(n\) ) is the count of numbers from 1 to 100, which is 100.Understanding these components helps to apply the formula for the sum effectively. Every arithmetic sequence can be summed by using established series formulas, making it a useful tool in mathematics.
Series Formula
The series formula is a mathematical formula used to find the sum of sequential terms in arithmetic series effectively. For an arithmetic sequence, this formula is\[ S = \frac{n}{2} [a + l] \]where
  • \(S\) stands for the sum of the series.
  • \(n\) represents the number of terms in the sequence.
  • \(a\) is the first term of the series.
  • \(l\) is the last term of the series.
By substituting these values into the formula, you can calculate the sum directly. As shown in the exercise example, substituting \(n = 100\), \(a = 1\), and \(l = 100\), yields \(S = 5050\). This formula is a cornerstone for solving problems involving arithmetic series and allows for swift computation without manual addition of each individual term.

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Most popular questions from this chapter

A mathematics exam consists of 10 multiple-choice questions and 5 open-ended problems in which all work must be shown. If an examinee must answer 8 of the multiple-choice questions and 3 of the open-ended problems, in how many ways can the questions and problems be chosen?

The group should select real-world situations where the Fundamental Counting Principle can be applied. These could involve the number of possible student ID numbers on your campus, the number of possible phone numbers in your community, the number of meal options at a local restaurant, the number of ways a person in the group can select outfits for class, the number of ways a condominium can be purchased in a nearby community, and so on. Once situations have been selected, group members should determine in how many ways each part of the task can be done. Group members will need to obtain menus, find out about telephone-digit requirements in the community, count shirts, pants, shoes in closets, visit condominium sales offices, and so on. Once the group reassembles, apply the Fundamental Counting Principle to determine the number of available options in each situation. Because these numbers may be quite large, use a calculator.

Many graphing utilities have a sequence-graphing mode that plots. the terms of a sequence as points on a rectangular coordinate system. Consult your manual, if your graphing utility has this capability, use it to graph each of the sequences. What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}-\frac{2 n^{2}+5 n-7}{n^{3}} n:[0,10,1] \text { by } a_{n}:[0,2,0.2]$$

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is even or greater than \(3 ?\)

Suppose that it is a week in which the cash prize in Florida's LOTTO is promised to exceed \(\$ 50\) million. If a person purchases \(22,957,480\) tickets in LOTTO at \(\$ 1\) per ticket (all possible combinations), isn't this a guarantee of winning the lottery? Because the probability in this situation is 1, what's wrong with doing this?
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