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Arithmetic Series Formula: How to Calculate the Sum of an Arithmetic Sequence
In the realm of mathematics, sequences play a crucial role in understanding patterns and relationships between numbers. An arithmetic sequence, also known as an arithmetic progression, is a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference. For instance, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.
When we talk about the sum of an arithmetic sequence, we are referring to an arithmetic series. This article will delve into the fascinating world of arithmetic series and explore the formula used to calculate their sum.
Understanding Arithmetic Sequences
Before we dive into the formula, let’s solidify our understanding of arithmetic sequences. Here are the key characteristics:
- Common Difference (d): The constant difference between any two consecutive terms. In the example above, d = 3.
- First Term (a): The starting value of the sequence. In the example above, a = 2.
- Number of Terms (n): The total number of terms in the sequence.
The Arithmetic Series Formula
The sum (S) of an arithmetic series can be calculated using the following formula:
S = (n/2) * [2a + (n-1)d]
Where:
- S = Sum of the arithmetic series
- n = Number of terms
- a = First term
- d = Common difference
Step-by-Step Example
Let’s consider the arithmetic sequence 2, 5, 8, 11, 14. We want to find the sum of this series.
1. **Identify the values:**
- a = 2 (First term)
- d = 3 (Common difference)
- n = 5 (Number of terms)
2. **Substitute the values into the formula:**
S = (5/2) * [2(2) + (5-1)3]
3. **Simplify the equation:**
S = (5/2) * [4 + 12]
S = (5/2) * 16
S = 40
Therefore, the sum of the arithmetic series 2, 5, 8, 11, 14 is 40.
Practice Questions
Here are some practice questions to test your understanding:
- Find the sum of the arithmetic series 1, 4, 7, 10, 13.
- Calculate the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, 15…
- An arithmetic series has a first term of 5 and a common difference of 2. If the sum of the series is 75, find the number of terms.
Understanding the arithmetic series formula is essential for solving various problems involving arithmetic sequences. By applying the formula and following the steps outlined above, you can efficiently calculate the sum of any arithmetic series.