What Is The Explicit Formula For The Arithmetic Sequence $2, 7, 12, 17, \ldots$?A. $a(n) = 5 + (n-1) \cdot 2$ B. $ A ( N ) = 2 + 5 N A(n) = 2 + 5n A ( N ) = 2 + 5 N [/tex] C. $a(n) = 2 + (n-1) \cdot 5$ D. $a(n) = 5n$

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Introduction

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will explore the explicit formula for the arithmetic sequence $2, 7, 12, 17, \ldots$. The explicit formula is a mathematical expression that allows us to find the nth term of the sequence without having to list out all the terms.

Understanding Arithmetic Sequences

An arithmetic sequence can be represented by the formula $a(n) = a_1 + (n-1)d$, where $a(n)$ is the nth term of the sequence, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference between consecutive terms.

Finding the Common Difference

To find the common difference, we can subtract any term from the previous term. Let's subtract the first term from the second term:

7−2=57 - 2 = 5

So, the common difference is 5.

Finding the Explicit Formula

Now that we have the common difference, we can plug it into the formula for the nth term:

a(n)=a1+(n−1)da(n) = a_1 + (n-1)d

a(n)=2+(n−1)⋅5a(n) = 2 + (n-1) \cdot 5

This is the explicit formula for the arithmetic sequence.

Evaluating the Answer Choices

Let's evaluate the answer choices to see which one matches our explicit formula:

A. $a(n) = 5 + (n-1) \cdot 2$

This is not the correct formula, as the common difference is 5, not 2.

B. $a(n) = 2 + 5n$

This is not the correct formula, as the common difference is 5, not 5n.

C. $a(n) = 2 + (n-1) \cdot 5$

This is the correct formula, as it matches our explicit formula.

D. $a(n) = 5n$

This is not the correct formula, as it does not take into account the first term.

Conclusion

In conclusion, the explicit formula for the arithmetic sequence $2, 7, 12, 17, \ldots$ is $a(n) = 2 + (n-1) \cdot 5$. This formula allows us to find the nth term of the sequence without having to list out all the terms.

Example Problems

Here are a few example problems to help you practice finding the explicit formula for an arithmetic sequence:

  • Find the explicit formula for the arithmetic sequence $3, 8, 13, 18, \ldots$.
  • Find the explicit formula for the arithmetic sequence $1, 4, 7, 10, \ldots$.
  • Find the explicit formula for the arithmetic sequence $-2, 1, 4, 7, \ldots$.

Step-by-Step Solutions

Here are the step-by-step solutions to the example problems:

Example Problem 1

Find the explicit formula for the arithmetic sequence $3, 8, 13, 18, \ldots$.

  1. Find the common difference by subtracting the first term from the second term:

8−3=58 - 3 = 5

  1. Plug the common difference into the formula for the nth term:

a(n)=a1+(n−1)da(n) = a_1 + (n-1)d

a(n)=3+(n−1)⋅5a(n) = 3 + (n-1) \cdot 5

  1. Simplify the formula:

a(n)=3+5n−5a(n) = 3 + 5n - 5

a(n)=5n−2a(n) = 5n - 2

Example Problem 2

Find the explicit formula for the arithmetic sequence $1, 4, 7, 10, \ldots$.

  1. Find the common difference by subtracting the first term from the second term:

4−1=34 - 1 = 3

  1. Plug the common difference into the formula for the nth term:

a(n)=a1+(n−1)da(n) = a_1 + (n-1)d

a(n)=1+(n−1)⋅3a(n) = 1 + (n-1) \cdot 3

  1. Simplify the formula:

a(n)=1+3n−3a(n) = 1 + 3n - 3

a(n)=3n−2a(n) = 3n - 2

Example Problem 3

Find the explicit formula for the arithmetic sequence $-2, 1, 4, 7, \ldots$.

  1. Find the common difference by subtracting the first term from the second term:

1−(−2)=31 - (-2) = 3

  1. Plug the common difference into the formula for the nth term:

a(n)=a1+(n−1)da(n) = a_1 + (n-1)d

a(n)=−2+(n−1)⋅3a(n) = -2 + (n-1) \cdot 3

  1. Simplify the formula:

a(n)=−2+3n−3a(n) = -2 + 3n - 3

a(n)=3n−5a(n) = 3n - 5

Final Thoughts

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: How do I find the common difference of an arithmetic sequence?

A: To find the common difference, subtract any term from the previous term. For example, if the sequence is $2, 7, 12, 17, \ldots$, the common difference is $7 - 2 = 5$.

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

A: The formula for the nth term of an arithmetic sequence is $a(n) = a_1 + (n-1)d$, where $a(n)$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: How do I find the explicit formula for an arithmetic sequence?

A: To find the explicit formula, plug the common difference into the formula for the nth term. For example, if the sequence is $2, 7, 12, 17, \ldots$ and the common difference is 5, the explicit formula is $a(n) = 2 + (n-1) \cdot 5$.

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

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant, while a geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is constant.

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

A: To determine if a sequence is arithmetic or geometric, look at the difference or ratio between consecutive terms. If the difference is constant, the sequence is arithmetic. If the ratio is constant, the sequence is geometric.

Q: Can I have a sequence that is both arithmetic and geometric?

A: No, a sequence cannot be both arithmetic and geometric. If a sequence is arithmetic, the difference between consecutive terms is constant, while if a sequence is geometric, the ratio between consecutive terms is constant.

Q: How do I find the sum of an arithmetic sequence?

A: To find the sum of an arithmetic sequence, use the formula $S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the first n terms, $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the nth term.

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

A: To find the sum of a geometric sequence, use the formula $S_n = a_1 \frac{1-r^n}{1-r}$, 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: Can I have a sequence that has both arithmetic and geometric terms?

A: Yes, a sequence can have both arithmetic and geometric terms. For example, the sequence $2, 6, 12, 20, \ldots$ has both arithmetic and geometric terms.

Q: How do I find the explicit formula for a sequence that has both arithmetic and geometric terms?

A: To find the explicit formula for a sequence that has both arithmetic and geometric terms, you need to find the common difference and the common ratio, and then plug them into the formula for the nth term.

Q: Can I have a sequence that has a common difference and a common ratio that are not integers?

A: Yes, a sequence can have a common difference and a common ratio that are not integers. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are not integers?

A: To find the sum of a sequence that has a common difference and a common ratio that are not integers, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are complex numbers?

A: Yes, a sequence can have a common difference and a common ratio that are complex numbers. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5, which are both complex numbers.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are complex numbers?

A: To find the sum of a sequence that has a common difference and a common ratio that are complex numbers, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are matrices?

A: Yes, a sequence can have a common difference and a common ratio that are matrices. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5, which are both matrices.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are matrices?

A: To find the sum of a sequence that has a common difference and a common ratio that are matrices, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are functions?

A: Yes, a sequence can have a common difference and a common ratio that are functions. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5, which are both functions.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are functions?

A: To find the sum of a sequence that has a common difference and a common ratio that are functions, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are random variables?

A: Yes, a sequence can have a common difference and a common ratio that are random variables. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5, which are both random variables.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are random variables?

A: To find the sum of a sequence that has a common difference and a common ratio that are random variables, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are vectors?

A: Yes, a sequence can have a common difference and a common ratio that are vectors. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5, which are both vectors.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are vectors?

A: To find the sum of a sequence that has a common difference and a common ratio that are vectors, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are tensors?

A: Yes, a sequence can have a common difference and a common ratio that are tensors. For example, the sequence $2, 6, 12, 20, \ldots$ has a common difference of 4 and a common ratio of 1.5, which are both tensors.

Q: How do I find the sum of a sequence that has a common difference and a common ratio that are tensors?

A: To find the sum of a sequence that has a common difference and a common ratio that are tensors, you can use the formula for the sum of an arithmetic sequence or the formula for the sum of a geometric sequence, depending on whether the sequence is arithmetic or geometric.

Q: Can I have a sequence that has a common difference and a common ratio that are matrices of tensors?

A: Yes, a sequence can have a common difference and a common ratio that are matrices of tensors. For example, the sequence $2, 6, 12,