Simplifying the nth Term Rule

Oct 13, 2025 by windailys.com 21 views

Introduction

In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers that follow a specific rule or pattern. In this article, we will explore a number grid and find the sequence rule by reading down a column of numbers, starting from 55.

The Number Grid

The number grid is a table of numbers arranged in rows and columns. The grid is shown below:

1	2	3	4	5	6	7	8
55	56	57	58	59	60	61	62
54	55	56	57	58	59	60	61
53	54	55	56	57	58	59	60
52	53	54	55	56	57	58	59
51	52	53	54	55	56	57	58
50	51	52	53	54	55	56	57
49	50	51	52	53	54	55	56
48	49	50	51	52	53	54	55

Finding the Sequence Rule

To find the sequence rule, we need to read down a column of numbers, starting from 55. Let's take the first column:

55, 54, 53, 52, 51, 50, 49, 48

We can see that the numbers are decreasing by 1 each time. This is a simple arithmetic sequence with a common difference of -1.

The nth Term Rule

Now that we have found the sequence rule, we can write the nth term rule for the sequence. The nth term rule is a formula that gives us the nth term of the sequence.

Let's denote the nth term as an. We can write the nth term rule as:

an = 55 - (n - 1)

where n is the term number.

Simplifying the nth Term Rule

We can simplify the nth term rule by expanding the expression:

an = 55 - (n - 1) an = 55 - n + 1 an = 56 - n

Conclusion

In this article, we explored a number grid and found the sequence rule by reading down a column of numbers, starting from 55. We discovered that the sequence is an arithmetic sequence with a common difference of -1. We then wrote the nth term rule for the sequence and simplified it to an = 56 - n.

Example Problems

Here are some example problems to test your understanding of the sequence rule:

Find the 5th term of the sequence.
Find the 10th term of the sequence.
Find the 15th term of the sequence.

Solution to Example Problems

To find the 5th term, we can plug in n = 5 into the nth term rule:

a5 = 56 - 5 a5 = 51

To find the 10th term, we can plug in n = 10 into the nth term rule:

a10 = 56 - 10 a10 = 46

To find the 15th term, we can plug in n = 15 into the nth term rule:

a15 = 56 - 15 a15 = 41

Final Thoughts

In conclusion, finding the sequence rule is an essential skill in mathematics. By understanding the sequence rule, we can solve problems and make predictions about the behavior of the sequence. We hope that this article has helped you understand the sequence rule and how to apply it to solve problems.

Glossary of Terms

Arithmetic sequence: A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
Common difference: The fixed constant that is added to each term to obtain the next term.
Nth term rule: A formula that gives us the nth term of a sequence.
Sequence: A list of numbers that follow a specific rule or pattern.

References

[1] "Sequences and Series" by Math Open Reference
[2] "Arithmetic Sequences and Series" by Khan Academy

About the Author

Introduction

In our previous article, we explored a number grid and found the sequence rule by reading down a column of numbers, starting from 55. We discovered that the sequence is an arithmetic sequence with a common difference of -1. We then wrote the nth term rule for the sequence and simplified it to an = 56 - n.

In this article, we will answer some frequently asked questions about sequences and series. We will cover topics such as what is a sequence, what is a series, and how to find the nth term rule.

Q: What is a sequence?

A: A sequence is a list of numbers that follow a specific rule or pattern. Sequences can be finite or infinite, and they can be represented in various ways, such as in a table, a graph, or a formula.

Q: What is a series?

A: A series is the sum of the terms of a sequence. Series can be finite or infinite, and they can be represented in various ways, such as in a formula or a graph.

Q: How do I find the nth term rule for a sequence?

A: To find the nth term rule for a sequence, you need to identify the pattern or rule that the sequence follows. You can do this by examining the sequence and looking for a common difference, ratio, or other pattern. Once you have identified the pattern, you can write the nth term rule as a formula.

Q: What is the difference between an arithmetic sequence and a geometric sequence?

A: An arithmetic sequence is a sequence in which each term is obtained by adding a fixed constant to the previous term. A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant.

Q: How do I find the sum of a finite series?

A: To find the sum of a finite series, you can use the formula for the sum of a finite arithmetic series:

S = (n/2)(a + l)

where S is the sum, n is the number of terms, a is the first term, and l is the last term.

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

A: To find the sum of an infinite series, you need to use a different approach. One way to do this is to use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where S is the sum, a is the first term, and r is the common ratio.

Q: What is the formula for the nth term of a geometric sequence?

A: The formula for the nth term of a geometric sequence is:

an = ar^(n-1)

where an is the nth term, a is the first term, r is the common ratio, and n is the term number.

Q: How do I find the common ratio of a geometric sequence?

A: To find the common ratio of a geometric sequence, you can use the formula:

r = (an / an-1)

where r is the common ratio, an is the nth term, and an-1 is the (n-1)th term.

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

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

S = a(1 - r^n) / (1 - r)

where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.

Conclusion

In this article, we answered some frequently asked questions about sequences and series. We covered topics such as what is a sequence, what is a series, and how to find the nth term rule. We also discussed the difference between an arithmetic sequence and a geometric sequence, and how to find the sum of a finite and infinite series.

Glossary of Terms

Arithmetic sequence: A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
Common difference: The fixed constant that is added to each term to obtain the next term.
Geometric sequence: A sequence of numbers in which each term is obtained by multiplying the previous term by a fixed constant.
Common ratio: The fixed constant that is multiplied by each term to obtain the next term.
Series: The sum of the terms of a sequence.
Nth term rule: A formula that gives us the nth term of a sequence.

References

[1] "Sequences and Series" by Math Open Reference
[2] "Arithmetic Sequences and Series" by Khan Academy
[3] "Geometric Sequences and Series" by Mathway

About the Author

The author is a mathematics enthusiast who loves to explore and explain mathematical concepts in an easy-to-understand way. If you have any questions or comments, please feel free to contact us.

1	2	3	4	5	6	7	8
55	56	57	58	59	60	61	62
54	55	56	57	58	59	60	61
53	54	55	56	57	58	59	60
52	53	54	55	56	57	58	59
51	52	53	54	55	56	57	58
50	51	52	53	54	55	56	57
49	50	51	52	53	54	55	56
48	49	50	51	52	53	54	55

1	2	3	4	5	6	7	8
55	56	57	58	59	60	61	62
54	55	56	57	58	59	60	61
53	54	55	56	57	58	59	60
52	53	54	55	56	57	58	59
51	52	53	54	55	56	57	58
50	51	52	53	54	55	56	57
49	50	51	52	53	54	55	56
48	49	50	51	52	53	54	55

1	2	3	4	5	6	7	8
55	56	57	58	59	60	61	62
54	55	56	57	58	59	60	61
53	54	55	56	57	58	59	60
52	53	54	55	56	57	58	59
51	52	53	54	55	56	57	58
50	51	52	53	54	55	56	57
49	50	51	52	53	54	55	56
48	49	50	51	52	53	54	55