The Second Term of an Arithmetic Sequence

Oct 13, 2025 by talynt.org 31 views

**Understanding Arithmetic Sequences: A Comprehensive Guide** ===========================================================

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. Arithmetic sequences are an essential concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics.

The Second Term of an Arithmetic Sequence

The second term of an arithmetic sequence is the term that comes after the first term. In this article, we are given that the second term of an arithmetic sequence has a value of 20.

The Fifth Term of an Arithmetic Sequence

The fifth term of an arithmetic sequence is the term that comes after the fourth term. In this article, we are given that the fifth term of an arithmetic sequence has a value of 38.

Determining the Constant Difference

To determine the constant difference for this arithmetic sequence, we can use 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.

We are given that the second term (a_2) is 20 and the fifth term (a_5) is 38. We can use this information to set up two equations:

a_2 = a_1 + d 20 = a_1 + d

a_5 = a_1 + 4d 38 = a_1 + 4d

We can solve these equations simultaneously to find the value of d.

Solving for d

Subtracting the first equation from the second equation, we get:

18 = 3d d = 6

Therefore, the constant difference for this arithmetic sequence is 6.

Determining the First Term

Now that we have found the value of d, we can substitute it into one of the original equations to find the value of a_1. We will use the first equation:

20 = a_1 + d 20 = a_1 + 6 a_1 = 14

Therefore, the first term of this arithmetic sequence is 14.

Determining a Formula for the nth Term

Now that we have found the values of a_1 and d, we can use the formula:

a_n = a_1 + (n-1)d

to determine a formula for the nth term of this arithmetic sequence.

a_n = 14 + (n-1)6 a_n = 14 + 6n - 6 a_n = 8 + 6n

Therefore, the formula for the nth term of this arithmetic sequence is a_n = 8 + 6n.

Conclusion

In this article, we have determined the constant difference for an arithmetic sequence, found the first term of the sequence, and determined a formula for the nth term of the sequence. We have used the given information about the second and fifth terms of the sequence to solve for the common difference and the first term. We have also used the formula for the nth term of an arithmetic sequence to determine a formula for the nth term of this specific sequence.

Frequently Asked Questions

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: What is the common difference?

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.

Q: How do I determine the common difference?

A: To determine the common difference, you can use 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.

Q: How do I determine the first term of an arithmetic sequence?

A: To determine the first term of an arithmetic sequence, you can use 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.

Q: How do I determine a formula for the nth term of an arithmetic sequence?

A: To determine a formula for the nth term of an arithmetic sequence, you can use 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.

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 of the sequence, a_1 is the first term, n is the term number, and d is the common difference.