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Chapter 2: Problem 36

\(\varnothing\) \(\\{101,102,103, \ldots, 200\\}\)

Short Answer

Expert verified
The sum of the sequence 101 to 200 is 15050.

Step by step solution

01

Identify the sequence

Looking at the numbers given, they start at 101 and increase by 1 each time, up to 200. This is an arithmetic sequence with a common difference of 1.
02

Find the formula for the sum of an arithmetic sequence

The sum \(S\) of an arithmetic series is given by the formula \(S = \frac{n}{2} (a+l)\), where \(n\) is the number of terms, \(a\) is the first term and \(l\) is the last term.
03

Calculate the sum of the series

Now, calculate the sum using the formula. Here, \(a = 101\), \(l = 200\), and \(n = 200 - 101 + 1 = 100\). So, \(S = \frac{100}{2} (101 + 200) = 50 * 301 = 15050.\)

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Key Concepts

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

Sum of an Arithmetic Series
In mathematics, an arithmetic series is simply the sum of an arithmetic sequence. An arithmetic sequence is a list of numbers with a pattern鈥攕pecifically, each number after the first is the result of adding a fixed number, called the common difference, to the previous term.
To find the sum of an arithmetic series, you use the formula:
    \[ S = \frac{n}{2} \times (a + l) \]
Here's what each part of the formula means:
  • \(S\): The sum of the arithmetic series.
  • \(n\): The total number of terms in the series.
  • \(a\): The first term of the series.
  • \(l\): The last or final term of the series.
This handy formula helps you quickly find the sum without adding each number one at a time, which can be very useful when dealing with large sequences like the one from 101 to 200.
Common Difference
The common difference in an arithmetic sequence is a crucial concept that allows an arithmetic pattern to persist through each term. It's the "fixed number" that gets added to each term. For the sequence \(101, 102, 103, \ldots, 200\), the common difference is simply 1. This is because every number is just 1 more than the previous one.
When you identify the common difference:
  • You ensure the sequence is indeed arithmetic.
  • You can use this information to predict future terms.
  • It's also used in calculating the sum of the series when you're using specific formulas.
Thus, understanding the common difference is vital not only for recognizing arithmetic sequences but also for various calculations involving them.
Number of Terms
The number of terms in an arithmetic sequence is essential for calculating other elements of the sequence or series, like its sum. Using the formula to count terms isn't difficult:
    \[ n = \frac{l - a}{d} + 1 \]
Where:
  • \(n\) is the number of terms.
  • \(l\) is the last term.
  • \(a\) is the first term.
  • \(d\) is the common difference.
In the sequence ranging from 101 to 200, using this formula helps us find that there are 100 terms: \(n = \frac{200 - 101}{1} + 1 = 100\).
Knowing the number of terms directly affects calculations like the sum and can help in identifying the overall structure and length of the sequence, which is crucial when dealing with arithmetic sequences.

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Most popular questions from this chapter

In Exercises 1-12, let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} . \end{aligned} $$ Find each of the following sets. \((A \cap B \cap C)^{\prime}\)

This activity is a group research project and should result in a presentation made by group members to the entire class. Georg Cantor was certainly not the only genius in history who faced criticism during his lifetime, only to have his work acclaimed as a masterpiece after his death. Describe the life and work of three other people, including at least one mathematician, who faced similar circumstances.

The set of students who scored \(90 \%\) or above on exam 1 and not on exam 2

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In Exercises 13-24, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\\B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((A \cap B) \cup(A \cap C)\)
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