Find The Sum Of The First 7 Terms Of The Series \[$ \frac{1}{2}, 1 \frac{1}{2}, 4 \frac{1}{2}, 13 \frac{1}{2} \$\].
Introduction
In mathematics, a series is the sum of the terms of a sequence. Series can be finite or infinite, and they can be used to represent various mathematical concepts, such as the sum of an infinite geometric series or the sum of a finite arithmetic series. In this article, we will focus on finding the sum of the first 7 terms of a given series.
Understanding the Series
The given series is: { \frac{1}{2}, 1 \frac{1}{2}, 4 \frac{1}{2}, 13 \frac{1}{2} $}$. To find the sum of the first 7 terms, we need to identify the pattern of the series. Upon closer inspection, we can see that each term is obtained by adding 1.5 to the previous term.
Identifying the Pattern
Let's analyze the series:
- The first term is { \frac{1}{2} $}$.
- The second term is { 1 \frac{1}{2} = \frac{3}{2} $}$.
- The third term is { 4 \frac{1}{2} = \frac{9}{2} $}$.
- The fourth term is { 13 \frac{1}{2} = \frac{27}{2} $}$.
We can see that each term is obtained by adding 1.5 to the previous term. This suggests that the series is an arithmetic series with a common difference of 1.5.
Finding the Sum of the Series
Now that we have identified the pattern of the series, we can use the formula for the sum of an arithmetic series to find the sum of the first 7 terms.
The formula for the sum of an arithmetic series is:
S_n = \frac{n}{2} (a_1 + a_n)
where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
In this case, we want to find the sum of the first 7 terms, so n = 7. The first term is { \frac{1}{2} $}$, and the 7th term is { \frac{1}{2} + 6(1.5) = \frac{1}{2} + 9 = \frac{19}{2} $}$.
Plugging these values into the formula, we get:
S_7 = \frac{7}{2} (\frac{1}{2} + \frac{19}{2})
Simplifying the expression, we get:
S_7 = \frac{7}{2} (\frac{20}{2})
S_7 = \frac{7}{2} (10)
S_7 = 35
Therefore, the sum of the first 7 terms of the series is 35.
Conclusion
In this article, we have shown how to find the sum of a series by identifying the pattern of the series and using the formula for the sum of an arithmetic series. We have also provided a step-by-step guide on how to find the sum of the first 7 terms of a given series.
Example Use Cases
The formula for the sum of an arithmetic series can be used in a variety of situations, such as:
- Finding the sum of a series of numbers that are added together in a specific pattern.
- Calculating the total cost of a series of items that are priced at different amounts.
- Determining the total amount of money that will be earned from a series of investments.
Tips and Tricks
When working with series, it's essential to identify the pattern of the series and use the correct formula to find the sum. Here are some tips and tricks to keep in mind:
- Always read the problem carefully and identify the pattern of the series.
- Use the correct formula for the sum of an arithmetic series.
- Plug in the values correctly and simplify the expression.
- Check your work by plugging in the values into the formula and simplifying the expression.
By following these tips and tricks, you can become proficient in finding the sum of a series and apply it to real-world situations.
Common Mistakes to Avoid
When working with series, it's easy to make mistakes. Here are some common mistakes to avoid:
- Not identifying the pattern of the series.
- Using the wrong formula for the sum of an arithmetic series.
- Plugging in the values incorrectly.
- Not simplifying the expression correctly.
By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.
Conclusion
Q: What is a series in mathematics?
A: A series is the sum of the terms of a sequence. Series can be finite or infinite, and they can be used to represent various mathematical concepts, such as the sum of an infinite geometric series or the sum of a finite arithmetic series.
Q: What is the difference between a series and a sequence?
A: A sequence is a list of numbers in a specific order, while a series is the sum of the terms of a sequence. For example, the sequence 1, 2, 3, 4, 5 is a list of numbers, while the series 1 + 2 + 3 + 4 + 5 is the sum of the terms of the sequence.
Q: How do I identify the pattern of a series?
A: To identify the pattern of a series, you need to examine the terms of the series and look for a common difference or ratio between consecutive terms. For example, if the series is 2, 5, 8, 11, 14, you can see that each term is obtained by adding 3 to the previous term.
Q: What is the formula for the sum of an arithmetic series?
A: The formula for the sum of an arithmetic series is:
S_n = \frac{n}{2} (a_1 + a_n)
where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.
Q: How do I find the sum of the first n terms of an arithmetic series?
A: To find the sum of the first n terms of an arithmetic series, you need to plug in the values of n, a_1, and a_n into the formula for the sum of an arithmetic series.
Q: What is the difference between an arithmetic series and a geometric series?
A: An arithmetic series is a series in which each term is obtained by adding a fixed constant to the previous term, while a geometric series is a series in which each term is obtained by multiplying the previous term by a fixed constant.
Q: How do I find the sum of an infinite geometric series?
A: To find the sum of an infinite geometric series, you need to use the formula:
S = \frac{a}{1 - r}
where S is the sum of the series, a is the first term, and r is the common ratio.
Q: What is the significance of finding the sum of a series?
A: Finding the sum of a series is significant because it can be used to solve a wide range of mathematical problems, such as calculating the total cost of a series of items, determining the total amount of money that will be earned from a series of investments, and finding the sum of an infinite geometric series.
Q: What are some common mistakes to avoid when finding the sum of a series?
A: Some common mistakes to avoid when finding the sum of a series include:
- Not identifying the pattern of the series.
- Using the wrong formula for the sum of an arithmetic series.
- Plugging in the values incorrectly.
- Not simplifying the expression correctly.
Q: How can I practice finding the sum of a series?
A: You can practice finding the sum of a series by working through a series of problems, such as finding the sum of the first n terms of an arithmetic series, finding the sum of an infinite geometric series, and solving real-world problems that involve series.
Q: What are some real-world applications of finding the sum of a series?
A: Some real-world applications of finding the sum of a series include:
- Calculating the total cost of a series of items.
- Determining the total amount of money that will be earned from a series of investments.
- Finding the sum of an infinite geometric series.
- Solving problems in finance, economics, and engineering that involve series.
By practicing finding the sum of a series and understanding the formulas and concepts involved, you can become proficient in solving a wide range of mathematical problems and apply your skills to real-world situations.