The sum of the first n terms of an arithmetic sequence is calculated using the formula: n/2 (2a + (n−1)d). This formula uses the first term of the sequence (a), the common difference between terms (d), and the number of terms (n) to determine the sum.
Understanding Arithmetic Sequences
An arithmetic sequence, or AP, is a series of numbers where the difference between consecutive terms is constant. This consistent difference is referred to as the common difference, denoted by 'd'. For example, in the sequence 2, 4, 6, 8, 10, the common difference is 2.
Formula for the Sum
The formula to find the sum of the first 'n' terms of an arithmetic sequence is:
- 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
- d is the common difference
Breakdown of the Formula
This formula is derived from the understanding that the terms in an arithmetic sequence increase or decrease linearly. The sum is essentially the average of the first and last terms multiplied by the number of terms. The term 2a + (n-1)d calculates the value of the last term in the sequence (nth term, denoted as 'l'), and then this is added to the first term (a). Since this is the sum of the first and last, you divide by 2 and then multiply by the number of terms which gives the sum of all n terms.
Example
Let's consider an arithmetic sequence: 3, 7, 11, 15, 19.
Here, a = 3 (the first term), d = 4 (the common difference), and if we want the sum of the first 5 terms, n = 5.
Plugging these values into the formula:
S5 = 5/2 [2(3) + (5 - 1)4]
S5 = 5/2 [6 + (4)4]
S5 = 5/2 [6 + 16]
S5 = 5/2 [22]
S5 = 5 * 11
S5 = 55
So, the sum of the first 5 terms of this arithmetic sequence is 55.
Key Components Explained:
- First Term (a): This is the initial value of the sequence.
- Common Difference (d): The constant value by which each successive term differs from the previous one.
- Number of Terms (n): The total count of terms you want to sum up in the sequence.
Why This Formula?
- The formula is efficient for large sequences, avoiding the need to individually add each term.
- It clearly shows the relationship between the first term, common difference, and number of terms in determining the sum.
- It is derived by pairing the first and last terms, the second and second-to-last terms, and so on, each pair resulting in the same sum.
Conclusion
In summary, to calculate the sum of the first 'n' terms of an arithmetic sequence, you utilize the formula n/2 (2a + (n−1)d), where 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference. This formula simplifies a potentially cumbersome calculation and reveals the core relationship between the different parameters of the sequence and its total sum.
