The sum of the odd integers from 1 to 99 (the last odd integer before 100) is 2500.
Here's how to calculate that:
Understanding Arithmetic Series
The odd integers from 1 to 99 form an arithmetic series: 1, 3, 5, 7, ... , 99. In an arithmetic series, the difference between consecutive terms is constant (in this case, the common difference is 2).
Formula for the Sum of an Arithmetic Series
The sum (S) of an arithmetic series is calculated as:
S = (n/2) * (a + l)
Where:
- n = number of terms
- a = first term
- l = last term
Applying the Formula
-
Identify the first term (a) and last term (l):
- a = 1
- l = 99
-
Determine the number of terms (n):
- The odd integers from 1 to 99 are: 1, 3, 5, ..., 99
- To find 'n', we can use the formula for the nth term of an arithmetic sequence: l = a + (n-1)d, where 'd' is the common difference.
- 99 = 1 + (n - 1) * 2
- 98 = (n - 1) * 2
- 49 = n - 1
- n = 50
-
Calculate the sum (S):
- S = (50/2) * (1 + 99)
- S = 25 * 100
- S = 2500
Therefore, the sum of the odd integers from 1 to 99 is 2500. Since the question asks "from 1 to 100," and 100 is even, we are summing the odd numbers from 1 up to and including 99.
