How do you find the sum of the first n natural numbers?

The sum of the first n natural numbers can be found using a specific formula derived from the properties of arithmetic sequences.

Understanding Natural Numbers and Arithmetic Sequences

Natural numbers are the counting numbers, starting with 1 (1, 2, 3, 4, and so on). When these numbers are listed in sequence, they form an arithmetic progression where the difference between any two consecutive numbers is constant (in this case, 1).

The Formula

According to the provided reference, the formula for the sum of the first n natural numbers is:

∑n1 = [n(n+1)/2]

This formula is a result of applying the arithmetic sequence sum formula.

How to Use the Formula

To calculate the sum of the first n natural numbers, follow these steps:

1. Identify 'n': Determine the value of n, which represents the number of natural numbers you want to sum. For example, if you want to sum the first 5 natural numbers, then n = 5.
2. Apply the formula: Substitute the value of n into the formula [n(n+1)/2].
3. Calculate the result: Perform the calculations to find the sum.

Examples

Here are a few examples to demonstrate the use of the formula:

• Example 1: Sum of the first 5 natural numbers

  • n = 5
  • Sum = [5(5+1)/2] = [5(6)/2] = 30/2 = 15
  • Therefore, 1 + 2 + 3 + 4 + 5 = 15

• Example 2: Sum of the first 10 natural numbers

  • n = 10
  • Sum = [10(10+1)/2] = [10(11)/2] = 110/2 = 55
  • Therefore, 1 + 2 + 3 + ... + 10 = 55

• Example 3: Sum of the first 100 natural numbers

  • n = 100
  • Sum = [100(100+1)/2] = [100(101)/2] = 10100/2 = 5050
  • Therefore, 1 + 2 + 3 + ... + 100 = 5050

Practical Insights

• This formula provides a quick and efficient way to find the sum of a large number of consecutive natural numbers without needing to add them up individually.
• It is a fundamental concept in mathematics and is used in many different areas, such as statistics and computer science.

Summary Table