So first we need to know how many odd numbers there are between 100 and 200...
100<2n-1<200 add 1 to all terms
101<2n<201 divide all terms by 2
50.5<n<100.5, since n must be an integer
51≤n≤100 so there are 100-51+1=50 odd numbers
Notice that 2n-1 forms an arithmetic sequence from 101 to 199 in this instance.
The sum of any arithmetic sequence is the average of the first and last terms times the number of terms, mathematically:
s=(a+b)(n/2), where a and b are the first and last terms and n is the number of terms, we saw earlier that there are 50 terms and that a=101 and b=199 so:
s=(101+199)(50/2)
s=7500
So the sum of all odd numbers between 100 and 200 is 7500.
