The Given Arithmetic Series

Oct 13, 2025 by talynt.org 75 views

Arithmetic series are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will delve into the world of arithmetic series and explore the explicit rule for the arithmetic sequence on which the given series is based.

What is an Arithmetic Series?

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. The general form of an arithmetic series is:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term, and 'd' is the common difference.

The Given Arithmetic Series

The given arithmetic series is:

12 + 18 + 24 + 30 + ... + 198

To find the explicit rule for the arithmetic sequence on which this series is based, we need to identify the first term and the common difference.

Identifying the First Term and Common Difference

The first term of the series is 12. To find the common difference, we can subtract the first term from the second term:

18 - 12 = 6

Therefore, the common difference is 6.

The Explicit Rule for the Arithmetic Sequence

The explicit rule for the arithmetic sequence on which the series is based is given by:

a_n = 12 + (n-1) × 6

where 'a_n' is the nth term of the sequence, and 'n' is the term number.

Deriving the Explicit Rule

To derive the explicit rule, we can use the formula for the nth term of an arithmetic sequence:

a_n = a + (n-1)d

where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.

In this case, the first term 'a' is 12, and the common difference 'd' is 6. Substituting these values into the formula, we get:

a_n = 12 + (n-1) × 6

Verifying the Explicit Rule

To verify the explicit rule, we can plug in different values of 'n' and check if the resulting terms match the given series.

For example, if we plug in 'n = 1', we get:

a_1 = 12 + (1-1) × 6 = 12 + 0 = 12

This matches the first term of the series.

If we plug in 'n = 2', we get:

a_2 = 12 + (2-1) × 6 = 12 + 6 = 18

This matches the second term of the series.

We can continue this process for different values of 'n' to verify that the explicit rule matches the given series.

Conclusion

In this article, we have explored the concept of arithmetic series and derived the explicit rule for the arithmetic sequence on which the given series is based. We have identified the first term and common difference of the series, and used the formula for the nth term of an arithmetic sequence to derive the explicit rule. We have also verified the explicit rule by plugging in different values of 'n' and checking if the resulting terms match the given series.

Frequently Asked Questions

Q: What is an arithmetic series?

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

Q: What is the common difference in an arithmetic series?

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

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

A: To find the explicit rule for an arithmetic sequence, you need to identify the first term and the common difference, and then use the formula for the nth term of an arithmetic sequence.

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 + (n-1)d

where 'a' is the first term, 'd' is the common difference, and 'n' is the term number.

Q: How do I verify the explicit rule for an arithmetic sequence?

A: To verify the explicit rule for an arithmetic sequence, you need to plug in different values of 'n' and check if the resulting terms match the given series.

Glossary

Arithmetic series: A sequence of numbers in which the difference between any two consecutive terms is constant.
Common difference: The constant difference between any two consecutive terms in an arithmetic series.
Explicit rule: A formula that describes the nth term of an arithmetic sequence.
Nth term: The term in an arithmetic sequence that corresponds to a given value of 'n'.
Term number: The position of a term in an arithmetic sequence.

References

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

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

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.

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

A: To find the sum of an arithmetic series, you need to use the formula:

S_n = n/2 * (a + l)

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

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is:

S_n = n/2 * (a + l)

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

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

A: To find the nth term of an arithmetic sequence, you need to use the formula:

a_n = a + (n-1)d

where a_n is the nth term, a 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 + (n-1)d

where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

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

A: To find the sum of the first n terms of an arithmetic sequence, you need to use the formula:

S_n = n/2 * (a + a_n)

where S_n is the sum of the first n terms, a is the first term, a_n is the nth term, and n is the term number.

Q: What is the formula for the sum of the first n terms of an arithmetic sequence?

A: The formula for the sum of the first n terms of an arithmetic sequence is:

S_n = n/2 * (a + a_n)

where S_n is the sum of the first n terms, a is the first term, a_n is the nth term, and n is the term number.

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

A: To find the sum of an infinite arithmetic series, you need to use the formula:

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 sum of an infinite arithmetic series?

A: The formula for the sum of an infinite arithmetic series is:

S = a / (1 - r)

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

Q: How do I find the common ratio of an infinite arithmetic series?

A: To find the common ratio of an infinite arithmetic series, you need to use the formula:

r = a_n / a

where r is the common ratio, a_n is the nth term, and a is the first term.

Q: What is the formula for the common ratio of an infinite arithmetic series?

A: The formula for the common ratio of an infinite arithmetic series is:

r = a_n / a

where r is the common ratio, a_n is the nth term, and a is the first term.

Glossary

Arithmetic series: A sequence of numbers in which the difference between any two consecutive terms is constant.
Arithmetic sequence: A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
Common difference: The constant difference between any two consecutive terms in an arithmetic sequence.
Common ratio: The constant ratio between any two consecutive terms in an infinite arithmetic series.
Nth term: The term in an arithmetic sequence that corresponds to a given value of 'n'.
Term number: The position of a term in an arithmetic sequence.

References

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