Question

What is sum of all odd numbers between 100 and 200

Answer 1

Odd numbers are of the form:

2n-1

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.