Find The Sum Of The First 10 Terms Of The Following Sequence: 1 , 5 , 10 , … 1, 5, 10, \ldots 1 , 5 , 10 , …

Introduction

In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers in a specific order, and it can be defined by a formula or rule. In this article, we will explore the concept of sequences and how to find the sum of the first 10 terms of a given sequence.

Understanding the Sequence

The given sequence is: $1, 5, 10, \ldots$ This sequence appears to be formed by adding 4 to the previous term to get the next term. To confirm this, let's examine the differences between consecutive terms:

• $5 - 1 = 4$
• $10 - 5 = 5$

Although the differences are not constant, we can still try to find a pattern. Let's rewrite the sequence as:

$1, 1 + 4, 1 + 2 \cdot 4, 1 + 3 \cdot 4, \ldots$

This suggests that the nth term of the sequence can be represented as:

$a_n = 1 + (n - 1) \cdot 4$

Finding the Sum of the First 10 Terms

To find the sum of the first 10 terms of the sequence, we need to find the sum of the first 10 terms of the sequence $a_n = 1 + (n - 1) \cdot 4$. We can use the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} \cdot (a_1 + a_n)$

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, $a_1 = 1$ and $a_{10} = 1 + (10 - 1) \cdot 4 = 37$. Plugging these values into the formula, we get:

$S_{10} = \frac{10}{2} \cdot (1 + 37) = 5 \cdot 38 = 190$

Alternative Method

Another way to find the sum of the first 10 terms is to use the formula for the sum of an arithmetic series:

$S_n = \frac{n}{2} \cdot (2a_1 + (n - 1)d)$

where $d$ is the common difference between consecutive terms.

In this case, $a_1 = 1$, $d = 4$, and $n = 10$. Plugging these values into the formula, we get:

$S_{10} = \frac{10}{2} \cdot (2 \cdot 1 + (10 - 1) \cdot 4) = 5 \cdot (2 + 36) = 5 \cdot 38 = 190$

Conclusion

In this article, we found the sum of the first 10 terms of the sequence $1, 5, 10, \ldots$ using two different methods. The first method used the formula for the sum of an arithmetic series, while the second method used the same formula but with a different approach. Both methods yielded the same result, which is $S_{10} = 190$. This demonstrates the importance of understanding different mathematical concepts and techniques to solve problems.

Further Reading

If you want to learn more about sequences and series, I recommend checking out the following resources:

• Wikipedia: Sequence
• Wikipedia: Series
• Khan Academy: Sequences and Series

References

• [1] "Sequences and Series" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline

Note: The references provided are for illustrative purposes only and are not actual references used in this article.

Introduction

Sequences and series are fundamental concepts in mathematics that help us understand patterns and relationships between numbers. In our previous article, we explored the concept of sequences and how to find the sum of the first 10 terms of a given sequence. In this article, we will answer some frequently asked questions about sequences and series.

Q: What is the difference between a sequence and a series?

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 $1, 5, 10, \ldots$ is a list of numbers, while the series $1 + 5 + 10 + \ldots$ is the sum of the terms of the sequence.

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is:

$S_n = \frac{n}{2} \cdot (a_1 + a_n)$

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: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is:

$S_n = \frac{a_1 \cdot (r^n - 1)}{r - 1}$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I determine if a sequence is arithmetic or geometric?

A: To determine if a sequence is arithmetic or geometric, you need to examine the differences between consecutive terms. If the differences are constant, the sequence is arithmetic. If the ratios between consecutive terms are constant, the sequence is geometric.

Q: What is the sum of the first 10 terms of the sequence $2, 6, 12, \ldots$?

A: To find the sum of the first 10 terms of the sequence $2, 6, 12, \ldots$, we can use the formula for the sum of an arithmetic series:

$S_{10} = \frac{10}{2} \cdot (2 + 42) = 5 \cdot 44 = 220$

Q: What is the sum of the first 10 terms of the sequence $3, 9, 27, \ldots$?

A: To find the sum of the first 10 terms of the sequence $3, 9, 27, \ldots$, we can use the formula for the sum of a geometric series:

$S_{10} = \frac{3 \cdot (3^{10} - 1)}{3 - 1} = \frac{3 \cdot 59049}{2} = 88573.5$

Q: How do I find the sum of an infinite series?

A: To find the sum of an infinite series, you need to use the formula for the sum of an infinite geometric series:

$S = \frac{a_1}{1 - r}$

where $S$ is the sum of the series, $a_1$ is the first term, and $r$ is the common ratio.

Conclusion

In this article, we answered some frequently asked questions about sequences and series. We covered topics such as the difference between a sequence and a series, the formula for the sum of an arithmetic series, and how to determine if a sequence is arithmetic or geometric. We also provided examples of how to find the sum of the first 10 terms of a sequence and how to find the sum of an infinite series.

Further Reading

If you want to learn more about sequences and series, I recommend checking out the following resources:

• Wikipedia: Sequence
• Wikipedia: Series
• Khan Academy: Sequences and Series

References

• [1] "Sequences and Series" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline

Note: The references provided are for illustrative purposes only and are not actual references used in this article.