The sum of all odd integers from 1 to 199 is 10,000.
To understand how this is calculated, consider the sequence of odd integers: 1, 3, 5, 7, ..., 199. This is an arithmetic progression where the first term (a) is 1, the common difference (d) is 2, and the last term (l) is 199.
First, we need to find the number of terms (n) in this sequence. We can use the formula for the nth term of an arithmetic progression:
l = a + (n - 1)d
Substituting the values:
199 = 1 + (n - 1)2
198 = (n - 1)2
99 = n - 1
n = 100
So, there are 100 odd integers from 1 to 199.
Next, we can find the sum of this arithmetic progression using the formula:
S = (n/2) * (a + l)
Substituting the values:
S = (100/2) (1 + 199)
S = 50 200
S = 10000
Therefore, the sum of all odd integers from 1 to 199 is 10,000.
