A finite arithmetic series is the sum of a finite number of terms in an arithmetic sequence.
In simpler terms, an arithmetic series is created when you add up the terms of an arithmetic sequence. An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. When you only add up a certain, limited number of these terms, the result is a finite arithmetic series.
Key Characteristics:
- Arithmetic Sequence Foundation: The series is based on an arithmetic sequence where each term is obtained by adding a constant value (the common difference) to the previous term.
- Finite Number of Terms: The summation stops after a specific number of terms, distinguishing it from an infinite arithmetic series.
- Summation: It represents the result of adding up all the terms within the finite arithmetic sequence.
Example:
Consider the arithmetic sequence: 2, 4, 6, 8, 10. The common difference (d) is 2.
A finite arithmetic series formed from this sequence could be the sum of the first three terms: 2 + 4 + 6 = 12. Therefore, 12 is a finite arithmetic series.
Formula for the Sum of a Finite Arithmetic Series:
The sum (Sn) of the first n terms of an arithmetic series can be calculated using the following formula:
Sn = (n/2) * [2a + (n-1)d]
Where:
- Sn is the sum of the first n terms
- n is the number of terms
- a is the first term of the sequence
- d is the common difference
Alternatively, if you know the first term (a) and the last term (l) of the sequence:
Sn = (n/2) * (a + l)
Example using the Formula:
Let's use the sequence 2, 4, 6, 8, 10 again and calculate the sum of the first 3 terms using the formula:
- n = 3
- a = 2
- d = 2
S3 = (3/2) [2(2) + (3-1)2]
S3 = (3/2) [4 + 4]
S3 = (3/2) * 8
S3 = 12
As we saw before by simply adding the first three terms, the sum is indeed 12.
In Conclusion:
A finite arithmetic series is the sum of a fixed number of terms taken from an arithmetic sequence, where each term increases (or decreases) by a constant amount. Formulas exist to easily calculate the sum of these series, making them easier to work with.
