The sum of all even numbers up to 100 is 2550. Here's how you can calculate it:
There are several ways to find this sum:
1. Using Arithmetic Progression (AP) Formula:
Even numbers up to 100 form an arithmetic progression: 2, 4, 6, ..., 100.
- First term (a): 2
- Common difference (d): 2
- Last term (l): 100
First, we need to find the number of terms (n) in the sequence. We can use the formula:
l = a + (n - 1)d
100 = 2 + (n - 1)2
98 = (n - 1)2
49 = n - 1
n = 50
Now, we can use the formula for the sum of an arithmetic progression:
S = (n/2) * (a + l)
S = (50/2) * (2 + 100)
S = 25 * 102
S = 2550
2. Using Formula for Sum of First n Even Numbers:
The sum of the first 'n' even numbers is given by the formula:
S = n(n + 1)
Since we want the sum of even numbers up to 100, and 100 is the 50th even number (100 / 2 = 50), we have n = 50.
S = 50 * (50 + 1)
S = 50 * 51
S = 2550
3. Manual Calculation (Less Efficient):
While not practical, you could theoretically add all the even numbers: 2 + 4 + 6 + ... + 98 + 100. However, this method is prone to errors and time-consuming.
Therefore, using either the arithmetic progression formula or the formula for the sum of the first n even numbers, we arrive at the same answer.
