Q7 of 67 Page 74

Find the sum of all the odd numbers divisible by 3 between 1 and 1000.

Starting from 1 the first odd number divisible by 3 is 3 then second odd number divisible by 3 is 9.
Greatest odd number till 1000 i.e. divisible by 3 is 999
Thus arithmetic progression is 3, 9, 15, …, 999
Here first term a = 3 and last term l = 999
and common difference = 9 – 3 = 6.


Let number of all odd number divisible by 3 be n.
Since, an = a + (n - 1)d


Where,


a = First term of AP
d = Common difference of AP
and no of terms is ‘n’



⇒ 999 = 3 + (n - 1) × 6
⇒ 996 = (n - 1) × 6
⇒ 166 = n – 1
⇒ n = 167


Since the sum of n terms is
Sn =
So sum of 167 terms S167
= 83.5 × [6 + 166 × 6]
=n
= 83.5 × 1002
= 83667


Hence, Sum is 83667.


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