Find the sum of the multiples of 3 less than 100. Hence or otherwise find the sum of the numbers less than 100 which are not multiples of 3.

Question

Sn = n/2( 2a +(n-1)d)

Answer 1

The multiples of three less than 100 are arithmetic sequence with n =33 terms.  The series sum of an arithmetic sequence is n( average of first and last term)  So for the multiples of three this is 33(3 + 99)/2 = 1683. 

The sum of the natural numbers less than 100 is, by the same formula,  99(1 + 99)/2 = 4950.  The desired sum is the difference 4950 - 1683 = 3267.