Calculate the Sum of Even Numbers
Choose a calculation, enter your numbers, and get an exact result instantly using an arithmetic formula rather than a long loop.
Formula for the Sum of Even Numbers
If you want the sum of all positive even numbers from 1 to n, first count how many even numbers there are. That count is:
The even numbers are then 2, 4, 6, ..., 2k . Factoring out 2 gives:
Since the sum of the first k positive integers is k(k + 1) / 2 , the 2 cancels the denominator:
Therefore, the main formula is floor(n / 2) × (floor(n / 2) + 1) . This gives the exact answer without adding every individual even number.
Sum of the First n Even Numbers
Do not confuse “even numbers up to n” with “the first n even numbers.” The first n even numbers are:
Their sum is especially simple:
For example, the first 10 even numbers end at 20, and their sum is 10 × 11 = 110.
Sum of Even Numbers Between Two Numbers
For a range such as 20 to 100, use the first even number at or above the lower endpoint and the last even number at or below the upper endpoint. The resulting sequence is an arithmetic progression with common difference 2.
Sum = number of terms × (first even + last even) / 2
Example: 20 + 22 + 24 + ... + 100 has 41 terms, so its sum is 41 × (20 + 100) / 2 = 2,460.
Examples of Sums of Even Numbers
| n | Even numbers | Count | Sum |
|---|---|---|---|
| 10 | 2 + 4 + 6 + 8 + 10 | 5 | 30 |
| 20 | 2 + 4 + ... + 20 | 10 | 110 |
| 50 | 2 + 4 + ... + 50 | 25 | 650 |
| 100 | 2 + 4 + ... + 100 | 50 | 2,550 |
| 1,000 | 2 + 4 + ... + 1,000 | 500 | 250,500 |
Sum of Even Numbers vs. Odd Numbers
Even and odd sums have closely related formulas. For even numbers, the sum of the first n terms is n(n + 1). For the first n odd numbers, the sum is n².
2 + 4 + 6 + ... + 2n
1 + 3 + 5 + ... + (2n − 1)
Why the Formula Works
The sequence 2, 4, 6, ..., 2k is an arithmetic sequence. Each term increases by exactly 2. Pairing the first and last terms gives the same total as pairing the second and second-to-last terms, and so on.
There are k terms, so the arithmetic-series formula gives:
This is why the calculator can handle large inputs efficiently: it uses the closed-form arithmetic formula instead of performing one addition for every even number.
Useful Properties of Even-Number Sums
- Number of positive evens up to n: floor(n / 2).
- Largest even number up to n: 2 × floor(n / 2).
- First n even numbers: 2, 4, 6, ..., 2n.
- Sum of first n even numbers: n(n + 1).
- Average of 2 through 2n: n + 1.
- The total is always even: every term is divisible by 2.
- For a range: use the arithmetic-series formula with common difference 2.
Frequently Asked Questions
What is the formula for the sum of even numbers?
If k = floor(n / 2) , the sum of the positive even numbers from 1 through n is k(k + 1) .
What is the sum of the first n even numbers?
The first n even numbers are 2, 4, 6, ..., 2n, and their sum is n(n + 1) .
How many even numbers are there from 1 to n?
There are floor(n / 2) positive even numbers from 1 through n.
What is the sum of even numbers up to 100?
There are 50 even numbers from 2 to 100. The sum is 50 × 51 = 2,550.
Is the sum of even numbers always even?
Yes. Since every even number is divisible by 2, adding any collection of even numbers produces another even number.
Does the calculator include 0?
The input may be 0, but the calculator defines “even numbers up to n” here as the positive even numbers 2, 4, 6, and so on. Therefore, for n = 0 the result is 0. Including 0 would not change the sum.
Can I calculate the sum between two numbers?
Yes. Select “Sum of even numbers between two numbers” and enter the lower and upper endpoints. Odd endpoints are handled automatically.
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