Answer:

The odd numbers between 1 and 1000 divisible by 3 are 3, 9, 15,…,999

Number of terms be ‘n’, so the nth term is 999

a = 3, d = 9-3 = 6, a_n = 999

a_n = a + (n-1)d

999 = 3 + (n-1)6

999 = 3 + 6n – 6

6n = 999 + 6 – 3

6n = 1002

n = 1002/6

n = 167

By using the formula,

Sum of n terms,

S = n/2 [a + l]

S = 167/2 [3 + 999]

S = 167/2 [1002]

S = 167 [501]

S = 83667

∴ The sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Thus, proved.