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1.3. Example 1
Interactive Audio Lesson
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Today, we will discuss arithmetic series. Can anyone tell me what an arithmetic series is?
I think it's a sequence of numbers where each number is added by a specific value.
Exactly! An arithmetic series is the sum of terms in an arithmetic sequence. What do we call the constant increase or decrease?
The common difference, right?
Precisely! Remember the acronym D.C. for 'Difference Constant' to help you recall this term. Now, who can give me the formula for the sum of the first n terms?
Is it S_n = n/2 [2a + (n - 1)d]?
Correct! We will use this formula in our next example.
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Let’s apply our formula to find the sum of the first ten terms of the series 3, 7, 11, 15... What is our first term?
The first term is 3.
Great! Now what about the common difference?
It's 4, because 7 minus 3 equals 4.
Exactly! Now let’s write the values we have: a = 3, d = 4, and n = 10. Who can substitute these into the formula?
S_{10} = 10/2 [2(3) + (10 - 1)(4)]
Awesome! Now, can you simplify this more?
Yes! S_{10} = 5[6 + 36] = 5 * 42 = 210.
Well done! The sum of the first ten terms is indeed 210.
Overview
Short Summary
This section illustrates finding the sum of the first ten terms of an arithmetic sequence, providing a concrete example for students.
Medium Summary
In this section, students learn how to apply the formula for the sum of an arithmetic series through a specific example. The example calculates the sum of the first ten terms of the arithmetic sequence starting with the term 3 and having a common difference of 4.
Detailed Summary
Example 1
In this portion of the chapter, we delve into an example that applies the concepts of arithmetic series. We start with the arithmetic sequence given by the terms 3, 7, 11, 15, etc. Here’s a breakdown of the example:
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Finding the sum of the first ten terms:
- First term (a): 3
- Common difference (d): 4
- Number of terms (n): 10
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Applying the formula:
- The sum (S) of the first n terms of an arithmetic series can be calculated using the formula:
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Calculating the sum:
- Plugging in the values, we compute:
The result, 210, shows the importance of understanding these series for real-life applications, setting a strong foundation for more advanced topics in mathematics.
Audio Book
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Create a free accountFind the sum of the first 10 terms of the arithmetic sequence: 3,7,11,15,…
Detailed Explanation
In this chunk, we are asked to find the sum of specific terms in an arithmetic sequence. The given sequence is: 3, 7, 11, 15, and continues with this pattern. To sum these terms, we first need to identify the properties of the arithmetic sequence:
- The first term (a) is the very first number in the sequence, which is 3.
- The common difference (d) is what we add to get from one term to the next. Here, we can see that if we take 7 - 3, we get 4, and similarly for other pairs, so d = 4.
- The number of terms (n) indicates how many terms we want to sum, which in this case is 10.
Examples & Analogies
Imagine you're collecting stamps, and every week, you find 4 more than you did the week before. If you started with 3 stamps, after 10 weeks, you'll want to know how many you've collected in total. This sequence represents your growing collection of stamps.
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Create a free accountSolution: • First term 𝑎 = 3 • Common difference 𝑑 = 4 • Number of terms 𝑛 = 10
10 𝑆 = [2(3)+(10−1)(4)] = 5[6+36] = 5×42 = 210 10 2
Detailed Explanation
Now we will use the formula for the sum of an arithmetic series to calculate the sum of the first 10 terms. The formula is:
S = n/2 * (2a + (n - 1)d)
Substituting the values:
- n = 10, a = 3, d = 4:
S = 10/2 * (2 * 3 + (10 - 1) * 4)
Calculating the equation step by step:
- Calculate 2 * 3 = 6.
- Calculate (10 - 1) * 4 = 36.
- Add these two results: 6 + 36 = 42.
- Multiply by 10/2 = 5: 5 * 42 = 210. Thus, the sum of the first 10 terms is 210.
Examples & Analogies
Think of this formula like a recipe for a cake. Each ingredient (number) is combined in a specific way to achieve the final product (total sum). Just like how measuring and mixing ingredients in the right amounts yields a delicious cake, plugging our numbers into the formula accurately gives us the correct sum.
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Arithmetic Series: The sum of terms of an arithmetic sequence.
Common Difference: The fixed interval between successive terms in an arithmetic sequence.
Sum Formula: S_n = n/2 [2a + (n - 1)d] is used for calculating the sum of an arithmetic series.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Stories
Memory Tools
Flash Cards
Glossary
Arithmetic Series
The sum of the terms in an arithmetic sequence.
Common Difference (d)
The fixed amount added or subtracted from each term in an arithmetic sequence.
First Term (a)
The initial term in an arithmetic sequence.
Number of Terms (n)
The total count of terms to be included in the sum of a series.