Find A Formula For The \[$ N \$\]th Term Of The Arithmetic Sequence.First Term: 12 Common Difference: 4 $\[ A_n = 4n + [?] \\]

Introduction

Arithmetic sequences are a fundamental concept in mathematics, and understanding how to find the formula for the nth term is crucial for solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will explore the process of finding the formula for the nth term of an arithmetic sequence, using the given first term and common difference as examples.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. For example, the sequence 12, 16, 20, 24, ... is an arithmetic sequence with a common difference of 4.

The Formula for the nth Term

The formula for the nth term of an arithmetic sequence is given by:

$$a_n = a_1 + (n-1)d$$

where:

• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence
• $n$ is the term number
• $d$ is the common difference

Finding the Formula for the nth Term

To find the formula for the nth term of an arithmetic sequence, we need to substitute the given values into the formula. Let's use the given first term (12) and common difference (4) as examples.

Step 1: Substitute the Given Values

Substitute the given values into the formula:

$$a_n = 12 + (n-1)4$$

Step 2: Simplify the Formula

Simplify the formula by distributing the 4:

$$a_n = 12 + 4n - 4$$

Step 3: Combine Like Terms

Combine like terms:

$$a_n = 8 + 4n$$

Conclusion

In this article, we have explored the process of finding the formula for the nth term of an arithmetic sequence. We have used the given first term and common difference as examples and have simplified the formula to obtain the final result. The formula for the nth term of an arithmetic sequence is given by:

$$a_n = a_1 + (n-1)d$$

where:

• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence
• $n$ is the term number
• $d$ is the common difference

Example Problems

Problem 1

Find the formula for the nth term of the arithmetic sequence with a first term of 5 and a common difference of 3.

Solution

Substitute the given values into the formula:

$$a_n = 5 + (n-1)3$$

Simplify the formula by distributing the 3:

$$a_n = 5 + 3n - 3$$

Combine like terms:

$$a_n = 2 + 3n$$

Problem 2

Find the formula for the nth term of the arithmetic sequence with a first term of 20 and a common difference of 2.

Solution

Substitute the given values into the formula:

$$a_n = 20 + (n-1)2$$

Simplify the formula by distributing the 2:

$$a_n = 20 + 2n - 2$$

Combine like terms:

$$a_n = 18 + 2n$$

Tips and Tricks

• Make sure to substitute the given values into the formula correctly.
• Simplify the formula by distributing the common difference.
• Combine like terms to obtain the final result.

Common Mistakes

• Failing to substitute the given values into the formula correctly.
• Not simplifying the formula by distributing the common difference.
• Not combining like terms to obtain the final result.

Conclusion

In conclusion, finding the formula for the nth term of an arithmetic sequence is a crucial concept in mathematics. By following the steps outlined in this article, you can easily find the formula for the nth term of an arithmetic sequence using the given first term and common difference. Remember to substitute the given values into the formula correctly, simplify the formula by distributing the common difference, and combine like terms to obtain the final result.

Introduction

In our previous article, we explored the process of finding the formula for the nth term of an arithmetic sequence. In this article, we will answer some frequently asked questions about the arithmetic sequence nth term formula.

Q&A

Q1: What is the formula for the nth term of an arithmetic sequence?

A1: The formula for the nth term of an arithmetic sequence is given by:

$$a_n = a_1 + (n-1)d$$

where:

• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence
• $n$ is the term number
• $d$ is the common difference

Q2: How do I find the formula for the nth term of an arithmetic sequence?

A2: To find the formula for the nth term of an arithmetic sequence, you need to substitute the given values into the formula and simplify it by distributing the common difference. Then, combine like terms to obtain the final result.

Q3: What is the common difference in an arithmetic sequence?

A3: The common difference in an arithmetic sequence is the constant difference between any two consecutive terms. For example, in the sequence 12, 16, 20, 24, ..., the common difference is 4.

Q4: How do I find the nth term of an arithmetic sequence if I know the first term and the common difference?

A4: To find the nth term of an arithmetic sequence, you can use the formula:

$$a_n = a_1 + (n-1)d$$

Substitute the given values into the formula and simplify it by distributing the common difference. Then, combine like terms to obtain the final result.

Q5: What is the formula for the nth term of an arithmetic sequence if the first term is 0?

A5: If the first term is 0, the formula for the nth term of an arithmetic sequence is given by:

$$a_n = (n-1)d$$

Q6: How do I find the formula for the nth term of an arithmetic sequence if the common difference is 0?

A6: If the common difference is 0, the formula for the nth term of an arithmetic sequence is given by:

$$a_n = a_1$$

Q7: Can I use the formula for the nth term of an arithmetic sequence to find the first term?

A7: Yes, you can use the formula for the nth term of an arithmetic sequence to find the first term. Rearrange the formula to solve for the first term:

$$a_1 = a_n - (n-1)d$$

Q8: How do I find the nth term of an arithmetic sequence if I know the first term and the sum of the first n terms?

A8: To find the nth term of an arithmetic sequence, you can use the formula:

$$a_n = \frac{1}{n} \left( \sum_{i=1}^{n} a_i - (n-1)a_1 \right)$$

where:

• $a_n$ is the nth term of the sequence
• $a_1$ is the first term of the sequence
• $n$ is the term number
• $\sum_{i=1}^{n} a_i$ is the sum of the first n terms

Conclusion

In conclusion, the arithmetic sequence nth term formula is a powerful tool for finding the nth term of an arithmetic sequence. By following the steps outlined in this article, you can easily find the formula for the nth term of an arithmetic sequence using the given first term and common difference. Remember to substitute the given values into the formula correctly, simplify the formula by distributing the common difference, and combine like terms to obtain the final result.