Question

What is the sum of all the even numbers from 0 to \(2 n,\) for any positive integer \(n ?\)

Short answer

The sum of all even numbers from 0 to \(2n\) is \(n(n + 1)\).

Step-by-step solution

Identify the Even Numbers

The even numbers in the range from 0 to \(2n\) are: \(0, 2, 4, ..., 2n\). We need to sum these numbers.

Recognize the Arithmetic Sequence

Observe that the even numbers form an arithmetic sequence with the first term (a) as 0, the common difference (d) as 2, and the last term as \(2n\).

Determine the Number of Terms

To find the number of terms in the sequence, use the formula for the \(k\)-th term of an arithmetic sequence: \(a_k = a + (k-1)d\). Here, the last term \(2n = 0 + (k-1) \times 2\). Hence, \(2n = (k-1) \times 2 \ => k - 1 = n \ => k = n + 1\). So, there are \(n + 1\) terms.

Use the Sum Formula for Arithmetic Sequence

The sum \(S\) of the first \(k\) terms of an arithmetic sequence is given by the formula \(S = \frac{k}{2} \times (a + l)\), where \(a\) is the first term, \(l\) is the last term. Here, \(k = n + 1\), \(a = 0\), and \(l = 2n\). \ Hence, \(S = \frac{n + 1}{2} \times (0 + 2n) \ = \frac{n + 1}{2} \times 2n \ = (n + 1) \times n\). \ Therefore, \(S = n(n + 1)\).

Key concepts

arithmetic sequences

An arithmetic sequence is a list of numbers where each number is obtained by adding a fixed value, called the common difference, to the previous number. For example, in the sequence 2, 5, 8, 11, ..., the common difference is 3 because each number is 3 more than the previous one.
Understanding the properties of arithmetic sequences helps in solving problems like summing even numbers.

sum of arithmetic series

The sum of an arithmetic series, which is just a fancy way of saying the sum of terms in an arithmetic sequence, can be found using a specific formula. If you have the first term (a), the last term (l), and the number of terms (k), you can use the formula: \[ S = \frac{k}{2} \times (a + l) \]
This formula tells you the sum (\textbf{S}) by multiplying the average of the first and last term by the number of terms, then dividing by 2.

even numbers

Even numbers are integers that are exactly divisible by 2, meaning there is no remainder when divided by 2. Examples include 0, 2, 4, 6, 8, and so on.
When looking at a range of even numbers, they naturally form an arithmetic sequence. For example, in the range from 0 to \(2n\), the even numbers are: 0, 2, 4, ..., \(2n\). The common difference (d) is 2.
Summing these numbers involves recognizing the arithmetic sequence and applying the sum formula for an arithmetic series.