The sum of the first n terms of an arithmetic progression (AP) with first term A and common difference D is given by a specific formula. This formula is derived from understanding the properties of an AP, where each term is obtained by adding the common difference to the previous term.
The Formula
The formula for the sum of the first n terms (Sn) of an arithmetic progression 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 of the arithmetic progression.
- D is the common difference between consecutive terms.
Explanation of the Formula
The formula can be understood as the average of the first and last term, multiplied by the number of terms. The term 2A + (n - 1)D represents twice the first term plus the common difference multiplied by (n-1). This effectively calculates the nth term of the sequence. Dividing by 2 gives the average of the first and last terms. Multiplying this average by the number of terms, n, yields the sum of the arithmetic progression. The reference provided confirms this formula: S_n=n/2(2a+(n-1)d)
Example
Let's consider an arithmetic progression with the first term A = 2 and the common difference D = 3. We want to find the sum of the first 5 terms (n = 5).
Using the formula:
S5 = 5/2 [2(2) + (5 - 1)3]
S5 = 5/2 [4 + (4)3]
S5 = 5/2 [4 + 12]
S5 = 5/2 [16]
S5 = 5 * 8
S5 = 40
Therefore, the sum of the first 5 terms of this arithmetic progression is 40.
Alternative Formula
Another way to represent the sum of an AP is:
Sn = n/2 (A + L)
Where:
- L is the last term of the AP (i.e., the nth term). L can be expressed as A + (n-1)D.
This formula is useful if you already know the last term of the arithmetic progression.
