Find A 30 A_{30} A 30 Given The Sequence: A N = A N − 1 + 4 A_n = A_{n-1} + 4 A N = A N − 1 + 4 A 1 = − 2 A_1 = -2 A 1 = − 2

Mar 13, 2025 by ADMIN 133 views

**Finding the 30th Term of a Sequence: A Step-by-Step Guide**

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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 each term is determined by a specific rule or formula. In this article, we will explore how to find the 30th term of a sequence given by the formula $a_n = a_{n-1} + 4$ , where $a_1 = -2$ . We will break down the problem into smaller steps and provide a clear explanation of each step.

Understanding the Sequence Formula

The given sequence formula is $a_n = a_{n-1} + 4$ . This means that each term in the sequence is obtained by adding 4 to the previous term. For example, if we want to find the second term, we add 4 to the first term, which is -2. So, the second term is $-2 + 4 = 2$ . Similarly, the third term is $2 + 4 = 6$ , and so on.

Finding the 30th Term

To find the 30th term, we need to apply the sequence formula 29 times, starting from the first term. We will use the formula $a_n = a_{n-1} + 4$ to find each term, and then use the result as the previous term for the next iteration.

Step 1: Finding the Second Term

We start by finding the second term using the formula $a_n = a_{n-1} + 4$ . We know that the first term is -2, so we add 4 to get the second term:

$a_2 = a_1 + 4 = -2 + 4 = 2$

Step 2: Finding the Third Term

Now that we have the second term, we can find the third term by adding 4 to the second term:

$a_3 = a_2 + 4 = 2 + 4 = 6$

Step 3: Finding the Fourth Term

We continue this process to find the fourth term by adding 4 to the third term:

$a_4 = a_3 + 4 = 6 + 4 = 10$

Step 4: Finding the 30th Term

We repeat this process 26 more times to find the 30th term. We can use a formula to simplify the process:

$a_n = a_1 + (n-1) \cdot 4$

We know that $a_1 = -2$ and $n = 30$ , so we can plug in these values to get:

$a_{30} = -2 + (30-1) \cdot 4$

$a_{30} = -2 + 29 \cdot 4$

$a_{30} = -2 + 116$

$a_{30} = 114$

Conclusion

In this article, we found the 30th term of a sequence given by the formula $a_n = a_{n-1} + 4$ , where $a_1 = -2$ . We broke down the problem into smaller steps and provided a clear explanation of each step. We used the sequence formula to find each term, and then used the result as the previous term for the next iteration. Finally, we used a formula to simplify the process and found the 30th term to be 114.

Example Use Cases

This problem can be applied to various real-world scenarios, such as:

Finance: Finding the future value of an investment based on a fixed interest rate.
Science: Modeling population growth or decay based on a fixed rate of change.
Engineering: Designing systems that require a specific sequence of events or actions.

Tips and Variations

Varying Interest Rates: If the interest rate changes over time, the sequence formula will need to be adjusted accordingly.
Non-Linear Sequences: If the sequence formula is non-linear, the problem will require a different approach.
Multiple Sequences: If there are multiple sequences involved, the problem will require a more complex solution.

Glossary

Sequence: A list of numbers in a specific order.
Term: A single number in a sequence.
Formula: A mathematical expression that describes a sequence.
Interest Rate: A fixed rate of change that is applied to a sequence.

References

Mathematics Textbook: A comprehensive textbook on mathematics that covers sequences and formulas.
Online Resources: Websites and online resources that provide information on sequences and formulas.
Mathematical Software: Software packages that can be used to calculate sequences and formulas.

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Introduction

In our previous article, we explored how to find the 30th term of a sequence given by the formula $a_n = a_{n-1} + 4$ , where $a_1 = -2$ . We provided a step-by-step guide on how to find the 30th term using the sequence formula. In this article, we will answer some frequently asked questions related to finding the 30th term of a sequence.

Q&A

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

A: The formula for finding the nth term of a sequence is $a_n = a_{n-1} + 4$ , where $a_1$ is the first term and $n$ is the term number.

Q: How do I find the 30th term of a sequence if I don't know the first term?

A: If you don't know the first term, you can use the formula $a_n = a_{n-1} + 4$ to find the 30th term. However, you will need to know the value of the 29th term to find the 30th term.

Q: Can I use a calculator to find the 30th term of a sequence?

A: Yes, you can use a calculator to find the 30th term of a sequence. Simply enter the formula $a_n = a_{n-1} + 4$ and the value of the first term, and the calculator will give you the value of the 30th term.

Q: What if the sequence formula is non-linear?

A: If the sequence formula is non-linear, you will need to use a different approach to find the 30th term. In this case, you may need to use a numerical method or a computer program to find the 30th term.

Q: Can I use the sequence formula to find the sum of the first n terms of a sequence?

A: Yes, you can use the sequence formula to find the sum of the first n terms of a sequence. However, you will need to use a different formula, such as the formula for the sum of an arithmetic series.

Q: What if I want to find the 30th term of a sequence that has a different formula?

A: If you want to find the 30th term of a sequence that has a different formula, you will need to use the new formula to find the 30th term. In this case, you may need to use a different approach or a computer program to find the 30th term.

Tips and Variations

Varying Interest Rates: If the interest rate changes over time, the sequence formula will need to be adjusted accordingly.
Non-Linear Sequences: If the sequence formula is non-linear, the problem will require a different approach.
Multiple Sequences: If there are multiple sequences involved, the problem will require a more complex solution.

Glossary

Sequence: A list of numbers in a specific order.
Term: A single number in a sequence.
Formula: A mathematical expression that describes a sequence.
Interest Rate: A fixed rate of change that is applied to a sequence.

References

Mathematics Textbook: A comprehensive textbook on mathematics that covers sequences and formulas.
Online Resources: Websites and online resources that provide information on sequences and formulas.
Mathematical Software: Software packages that can be used to calculate sequences and formulas.

Conclusion

In this article, we answered some frequently asked questions related to finding the 30th term of a sequence. We provided a step-by-step guide on how to find the 30th term using the sequence formula and discussed some common variations and tips. We hope that this article has been helpful in understanding how to find the 30th term of a sequence.