An arithmetic progression (AP) is a sequence of numbers in which each term is obtained by adding a constant difference (d) to the preceding term. In other words, an AP is a sequence of numbers in which each term differs from the preceding term by a fixed number called the common difference.
The formula for the nth term of an AP is given by:
an = a1 + (n-1)d
Where, an = nth term of the AP a1 = first term of the AP n = number of terms in the AP d = common difference
Examples of arithmetic progressions:
- 2, 4, 6, 8, 10, 12, …
In this AP, the first term (a1) is 2 and the common difference (d) is 2. The second term is obtained by adding 2 to the first term (a2 = a1 + d = 2 + 2 = 4). Similarly, the third term is obtained by adding 2 to the second term (a3 = a2 + d = 4 + 2 = 6), and so on.
- -3, 1, 5, 9, 13, …
In this AP, the first term (a1) is -3 and the common difference (d) is 4. The second term is obtained by adding 4 to the first term (a2 = a1 + d = -3 + 4 = 1). Similarly, the third term is obtained by adding 4 to the second term (a3 = a2 + d = 1 + 4 = 5), and so on.
- 10, 7, 4, 1, -2, …
In this AP, the first term (a1) is 10 and the common difference (d) is -3. The second term is obtained by adding -3 to the first term (a2 = a1 + d = 10 – 3 = 7). Similarly, the third term is obtained by adding -3 to the second term (a3 = a2 + d = 7 – 3 = 4), and so on.
Sum of N Terms in AP with Examples
The sum of the first n terms of an arithmetic progression (AP) is called the sum of n terms or the nth partial sum of the AP. The formula for finding the sum of the first n terms of an AP is given by:
Sn = (n/2)(a1 + an)
Where, Sn = sum of first n terms of the AP a1 = first term of the AP an = nth term of the AP
We can also write the formula for the nth term of an AP in terms of the common difference (d):
an = a1 + (n-1)d
Using this formula, we can rewrite the formula for the sum of n terms as:
Sn = (n/2)(2a1 + (n-1)d)
Examples:
- Find the sum of the first 10 terms of the AP 2, 4, 6, 8, …
Here, a1 = 2 and d = 2 Using the formula, Sn = (n/2)(2a1 + (n-1)d) = (10/2)(2(2) + (10-1)(2)) = 110 So, the sum of the first 10 terms of the AP is 110.
- Find the sum of the first 15 terms of the AP -3, 1, 5, 9, …
Here, a1 = -3 and d = 4 Using the formula, Sn = (n/2)(2a1 + (n-1)d) = (15/2)(2(-3) + (15-1)(4)) = 120 So, the sum of the first 15 terms of the AP is 120.
Note that the formula for the sum of n terms of an arithmetic progression can be used only when the number of terms is known. If the number of terms is not known, we can use other methods to find the sum, such as using the sum of the first and last terms, or using the sum of two APs with the same first term and common difference.