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

1.	Find the sum of all odd numbers, divisible by 3 between 1 and 1000.
Answer» Odd numbers divisible by 3, between 1 and 1000 are 3, 9, 15, 21 …….. 999. Clearly series 3, 9, 15, 15,21 …… 999 is A.P. whose first term (a) = 3 and common difference (d) = 6. Let us assume that this series contains n terms. ∴ a_n = 999 ⇒ a + (n – 1)d = 999 ⇒ 3 + (n – 1) × 6 = 999 ⇒ 6n – 3 = 999 ⇒ 6n = 1002 ⇒ n = 1002/6 ⇒ n = 167 ∴ Required sum S_n = n/2(a + l) S₁₆₇ = 167/2(3 + 999) = 167/2 × 1002 = 167 × 501 = 83667 Hence, required sum = 83667

Answer»

Odd numbers divisible by 3, between 1 and 1000 are 3, 9, 15, 21 …….. 999.

Clearly series 3, 9, 15, 15,21 …… 999 is A.P.

whose first term (a) = 3 and common difference (d) = 6.

Let us assume that this series contains n terms.

∴ a_n = 999

⇒ a + (n – 1)d = 999

⇒ 3 + (n – 1) × 6 = 999

⇒ 6n – 3 = 999

⇒ 6n = 1002

⇒ n = 1002/6

⇒ n = 167

∴ Required sum

S_n = n/2(a + l)

S₁₆₇ = 167/2(3 + 999)

= 167/2 × 1002

= 167 × 501

= 83667

Hence, required sum = 83667

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