An arithmetic series is simply the sum of the terms in an arithmetic sequence.
Understanding Arithmetic Sequences
Before diving into arithmetic series, it's helpful to understand arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.
- Example of an Arithmetic Sequence: 2, 4, 6, 8, 10... (The common difference is 2)
What is an Arithmetic Series?
An arithmetic series is the sum of the terms in an arithmetic sequence. So, if you have an arithmetic sequence, you can add up all the terms to get an arithmetic series.
- Example of an Arithmetic Series: The sum of the arithmetic sequence 2, 4, 6, 8, 10 is 2 + 4 + 6 + 8 + 10 = 30. Therefore, 30 is the arithmetic series for that sequence.
Formula for the Sum of an Arithmetic Series
There's a handy formula to calculate the sum (Sn) of an arithmetic series:
Sn = (n/2) * [2a + (n - 1)d]
Where:
- Sn = The sum of the first 'n' terms of the series
- n = The number of terms in the series
- a = The first term of the series
- d = The common difference
Alternatively, if you know the last term (l) of the sequence, you can use this simplified formula:
Sn = (n/2) * (a + l)
Example Calculation
Let's say we want to find the sum of the first 50 terms of the arithmetic sequence starting with 3 and having a common difference of 5.
Here, a = 3, d = 5, and n = 50.
Using the formula: Sn = (n/2) * [2a + (n - 1)d]
S50 = (50/2) * [2(3) + (50 - 1)5]
S50 = 25 * [6 + (49)5]
S50 = 25 * [6 + 245]
S50 = 25 * 251
S50 = 6275
Therefore, the sum of the first 50 terms of the arithmetic series is 6275.
Key Takeaways
- An arithmetic series is the sum of an arithmetic sequence.
- You can use formulas to quickly calculate the sum of an arithmetic series without manually adding all the terms.
- Identifying the first term, common difference, and number of terms is crucial for using the formulas.
