The sum of the first 10 natural numbers in an arithmetic progression is 55.
Understanding Arithmetic Progressions
An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. In the case of the first 10 natural numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10), the common difference is 1.
Calculating the Sum
There are a couple of ways to calculate the sum:
Method 1: Direct Addition
Simply add all the numbers together: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
Method 2: Using the Arithmetic Series Formula
The sum (S) of an arithmetic series can be calculated using the formula:
S = (n/2) * (a + l)
where:
- n = number of terms
- a = first term
- l = last term
In this case:
- n = 10
- a = 1
- l = 10
Therefore, S = (10/2) (1 + 10) = 5 11 = 55
Reference Confirmation
The provided reference states: "The sum of an A.P: 1, 2, 3, 4,5……. 10. S = 5[11] = 55. Therefore, the sum of the first ten natural numbers is 55." This confirms the answer obtained through both direct addition and the arithmetic series formula.
