The sum of an arithmetic series can be found using a specific formula that incorporates the first term, the common difference, and the number of terms. An arithmetic series is defined as a sequence of numbers where the difference between any two consecutive terms is constant.
Understanding the Components
To calculate the sum, you'll need three key pieces of information:
- a: The first term in the series.
- d: The common difference between consecutive terms.
- n: The number of terms in the series.
The Formula
The formula for the sum of an arithmetic series is:
Sum = n/2 [2a + (n - 1)d]
Where:
- 'n' is the number of terms
- 'a' is the first term
- 'd' is the common difference
Step-by-Step Calculation
Here's how to use the formula:
- Identify 'a': Determine the value of the first term in your series.
- Identify 'd': Calculate the common difference by subtracting any term from its subsequent term (e.g., the second term minus the first term).
- Identify 'n': Count how many terms there are in the series.
- Plug into the Formula: Substitute the values of 'a', 'd', and 'n' into the sum formula: n/2 [2a + (n - 1)d].
- Calculate: Perform the arithmetic operations to find the sum of the series.
Example
Let's say we have the series: 2, 4, 6, 8, 10.
- a = 2 (the first term)
- d = 2 (the common difference, 4 - 2 = 2)
- n = 5 (there are 5 terms)
Now, apply the formula:
Sum = 5/2 [2(2) + (5 - 1)2]
Sum = 2.5 [4 + (4)2]
Sum = 2.5 [4 + 8]
Sum = 2.5 [12]
Sum = 30
Therefore, the sum of the series 2, 4, 6, 8, 10 is 30.
Practical Applications
- Financial Calculations: Calculating the total value of investments growing at a constant rate over time.
- Construction: Estimating the total materials needed for a project when each stage requires an increasing amount.
- Physics: Determining the distance traveled by an object accelerating at a constant rate.
Key Takeaway
The formula n/2 [2a + (n - 1)d] provides a simple and efficient way to calculate the sum of an arithmetic series, which is applicable across various fields.
