Find The Sum Of The First 10 Terms Of This Series: 4, 8, 12,
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Introduction
In mathematics, a series is a sequence of numbers that are added together to find a total sum. The series can be finite or infinite, and it can be arithmetic, geometric, or a combination of both. In this article, we will focus on finding the sum of the first 10 terms of a given arithmetic series.
Understanding the Series
The given series is: 4, 8, 12, ...
This is an arithmetic series because each term is obtained by adding a fixed constant to the previous term. In this case, the common difference (d) is 4, which means each term is 4 more than the previous term.
Identifying the First Term and Common Difference
The first term (a) of the series is 4, and the common difference (d) is 4.
Finding the Sum of the First 10 Terms
To find the sum of the first 10 terms, we can use the formula for the sum of an arithmetic series:
S_n = n/2 * (a + l)
where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.
Calculating the Last Term
To find the last term (l), we can use the formula:
l = a + (n-1)d
where l is the last term, a is the first term, n is the number of terms, and d is the common difference.
Plugging in the values, we get:
l = 4 + (10-1)4 l = 4 + 36 l = 40
Finding the Sum
Now that we have the first term (a) and the last term (l), we can find the sum of the first 10 terms:
S_10 = 10/2 * (4 + 40) S_10 = 5 * 44 S_10 = 220
Conclusion
In this article, we found the sum of the first 10 terms of a given arithmetic series. We identified the first term and the common difference, calculated the last term, and used the formula for the sum of an arithmetic series to find the sum. The sum of the first 10 terms is 220.
Example Use Cases
This formula can be used to find the sum of any arithmetic series. For example, if we want to find the sum of the first 20 terms of the series 2, 6, 10, ..., we can use the same formula:
a = 2 d = 4 n = 20
l = a + (n-1)d l = 2 + (20-1)4 l = 2 + 76 l = 78
S_20 = 20/2 * (2 + 78) S_20 = 10 * 80 S_20 = 800
Tips and Variations
- To find the sum of an infinite arithmetic series, we can use the formula:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
- To find the sum of a geometric series, we can use the formula:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Conclusion
In conclusion, finding the sum of an arithmetic series is a straightforward process that involves identifying the first term and the common difference, calculating the last term, and using the formula for the sum of an arithmetic series. This formula can be used to find the sum of any arithmetic series, and it can be applied to a wide range of mathematical problems.
References
- "Arithmetic Series" by Math Open Reference
- "Sum of an Arithmetic Series" by Wolfram MathWorld
- "Arithmetic Series Formula" by Mathway
Further Reading
- "Arithmetic Progressions" by Khan Academy
- "Sum of an Arithmetic Series" by Brilliant
- "Arithmetic Series" by MIT OpenCourseWare
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Q: What is an arithmetic series?
A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.
Q: How do I find the sum of an arithmetic series?
A: To find the sum of an arithmetic series, you need to identify the first term (a), the common difference (d), and the number of terms (n). Then, you can use the formula:
S_n = n/2 * (a + l)
where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.
Q: What is the formula for the last term of an arithmetic series?
A: The formula for the last term (l) of an arithmetic series is:
l = a + (n-1)d
where l is the last term, a is the first term, n is the number of terms, and d is the common difference.
Q: Can I use the formula for the sum of an arithmetic series for an infinite series?
A: No, the formula for the sum of an arithmetic series is only applicable for finite series. If you want to find the sum of an infinite series, you need to use a different formula, such as:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: What is the difference between an arithmetic series and a geometric series?
A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric series, on the other hand, is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed constant, called the common ratio.
Q: Can I use the formula for the sum of an arithmetic series for a geometric series?
A: No, the formula for the sum of an arithmetic series is only applicable for arithmetic series. If you want to find the sum of a geometric series, you need to use a different formula, such as:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: How do I find the sum of the first n terms of a geometric series?
A: To find the sum of the first n terms of a geometric series, you need to identify the first term (a), the common ratio (r), and the number of terms (n). Then, you can use the formula:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Q: What is the formula for the sum of an infinite geometric series?
A: The formula for the sum of an infinite geometric series is:
S = a / (1 - r)
where S is the sum, a is the first term, and r is the common ratio.
Q: Can I use the formula for the sum of an infinite geometric series for a finite series?
A: No, the formula for the sum of an infinite geometric series is only applicable for infinite series. If you want to find the sum of a finite series, you need to use a different formula, such as:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Q: How do I determine if a series is arithmetic or geometric?
A: To determine if a series is arithmetic or geometric, you need to examine the relationship between consecutive terms. If the difference between consecutive terms is constant, the series is arithmetic. If each term is obtained by multiplying the previous term by a fixed constant, the series is geometric.
Q: Can I use the formula for the sum of an arithmetic series for a series with a negative common difference?
A: Yes, the formula for the sum of an arithmetic series is applicable for series with a negative common difference. However, you need to be careful when calculating the last term, as the formula may result in a negative value.
Q: Can I use the formula for the sum of a geometric series for a series with a negative common ratio?
A: Yes, the formula for the sum of a geometric series is applicable for series with a negative common ratio. However, you need to be careful when calculating the sum, as the formula may result in a negative value.
Q: How do I find the sum of a series with a variable common difference?
A: To find the sum of a series with a variable common difference, you need to use a different formula, such as:
S_n = n/2 * (a + l)
where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.
However, you need to be careful when calculating the last term, as the formula may result in a negative value.
Q: Can I use the formula for the sum of an arithmetic series for a series with a variable first term?
A: Yes, the formula for the sum of an arithmetic series is applicable for series with a variable first term. However, you need to be careful when calculating the last term, as the formula may result in a negative value.
Q: How do I find the sum of a series with a variable common ratio?
A: To find the sum of a series with a variable common ratio, you need to use a different formula, such as:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
However, you need to be careful when calculating the sum, as the formula may result in a negative value.
Q: Can I use the formula for the sum of a geometric series for a series with a variable first term?
A: Yes, the formula for the sum of a geometric series is applicable for series with a variable first term. However, you need to be careful when calculating the sum, as the formula may result in a negative value.
Q: How do I find the sum of a series with a variable number of terms?
A: To find the sum of a series with a variable number of terms, you need to use a different formula, such as:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
However, you need to be careful when calculating the sum, as the formula may result in a negative value.
Q: Can I use the formula for the sum of an arithmetic series for a series with a variable number of terms?
A: Yes, the formula for the sum of an arithmetic series is applicable for series with a variable number of terms. However, you need to be careful when calculating the last term, as the formula may result in a negative value.
Q: How do I find the sum of a series with a variable common difference and variable first term?
A: To find the sum of a series with a variable common difference and variable first term, you need to use a different formula, such as:
S_n = n/2 * (a + l)
where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.
However, you need to be careful when calculating the last term, as the formula may result in a negative value.
Q: Can I use the formula for the sum of a geometric series for a series with a variable common ratio and variable first term?
A: Yes, the formula for the sum of a geometric series is applicable for series with a variable common ratio and variable first term. However, you need to be careful when calculating the sum, as the formula may result in a negative value.
Q: How do I find the sum of a series with a variable common difference, variable first term, and variable number of terms?
A: To find the sum of a series with a variable common difference, variable first term, and variable number of terms, you need to use a different formula, such as:
S_n = a * (1 - r^n) / (1 - r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
However, you need to be careful when calculating the sum, as the formula may result in a negative value.
Q: Can I use the formula for the sum of an arithmetic series for a series with a variable common difference, variable first term, and variable number of terms?
A: Yes, the formula for the sum of an arithmetic series is applicable for series with a variable common difference, variable first term, and variable number of terms. However, you need to be careful when calculating the last term, as the