What is the Sum of the First n Even Numbers?

When working with sequences and series, it's important to understand the properties and formulas related to various types of number sequences, such as even numbers. This article explores the sum of the first n even numbers, providing formulas, examples, and a deeper understanding of arithmetic progressions.

Understanding Even Numbers and Arithmetic Progression

Even numbers are integers that are divisible by 2. The sequence of even numbers begins with 2 and increases by 2 for each subsequent even number: 2, 4, 6, 8, 10, and so on. This sequence is an arithmetic progression (AP) with the first term a 2 and a common difference d 2.

Formula for the Sum of the First n Even Numbers

Given the first term of an arithmetic progression (AP) and the common difference, the sum of the first n terms can be calculated using the formula:

Sn n/2[2a1 (n - 1)d]

For the sequence of even numbers, we substitute a1 2 and d 2:

Sn n/2[2 × 2 (n - 1) × 2] n/2[4 2n - 2] n/2[2n 2] n/2 × 2(n 1) n(n 1)

Thus, the sum of the first n even numbers is given by the formula:

Sn n2 n

Examples and Applications

Let's explore a few examples to illustrate the application of this formula:

Example 1: Sum of the First 10 Even Numbers

Using the formula, the sum of the first 10 even numbers (2, 4, 6, 8, 10, 12, 14, 16, 18, 20) is:

S10 102 10 100 10 110

This can also be calculated by adding the terms directly:

2 4 6 8 10 12 14 16 18 20 110

Example 2: Sum of the First 50 Even Numbers

Using the formula, the sum of the first 50 even numbers is:

S50 502 50 2500 50 2550

Example 3: Sum of the First 84 Even Numbers

The sum of the first 84 even numbers can be calculated as:

S84 842 84 7056 84 7140

Alternatively, this sum can also be found using the formula for the sum of an arithmetic series:

S84 42(1 2 3 ... 84) 42 × 84 × 85/2 7140

Conclusion

The sum of the first n even numbers is a fundamental concept in mathematics, applicable in various fields such as computer science, statistics, and engineering. Understanding the formula and its derivation can help in solving complex problems and in developing algorithms for various applications.

Additional Resources

For further reading and exercises, consider exploring the following resources:

Sum of the First n Integers and the Sum of the First n Even Numbers Math is Fun - Sum of Even Numbers Khan Academy - Sum of First n Even Numbers