according to given conditions the sequence is 3, 9, 15, 21….999

Given an AP whose first term is 3 and d is 6. Hence sum is given by

To prove: the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667

Hence the sum is given by the formula

Now for the finding number of terms, the formula is

n = 167

Substituting n is the sum formula we get

s = 83667

Hence, Proved.