 Questions & Answers    Question Answers

# Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.  Answer Verified
Hint: First, check whether the set of required numbers form a series and then find the total number of odd integers divisible by 3 between 1 and 1000. After that make a sum of that using the appropriate sum of series formula.

As, we all know that all odd numbers between 1 and 1000,
which are divisible by 3 are ${\text{3, 9, 15, }}......{\text{ 999}}$ which forms an A.P.
$\Rightarrow$First term of this A.P is ${a_1} = 3$.
$\Rightarrow$Second term of this A.P. is ${a_2} = 9$.
$\Rightarrow$Last term of this A.P. is ${a_n} = 999$.
$\Rightarrow$Common difference $d = {a_2} - {a_1} = 9 - 3 = 6$
So, we know that ${n^{th}}$ term of any A.P is given as
$\Rightarrow {a_n} = {a_1} + (n - 1)d$
For, finding the value of n.
On putting, ${a_n} = 999,{\text{ }}{a_1} = 3$ and $d = 6$ in the above equation. We get,
$\Rightarrow 999 = 3 + (n - 1)6 \\ \Rightarrow 999 = 3 + 6n - 6 \\$
On solving the above equation. We get,
$\Rightarrow n = \dfrac{{1002}}{6} = 167$ numbers in the A.P
Now, as we know that sum of these n terms of A.P is given by,
$\Rightarrow {S_n} = \dfrac{n}{2}[{a_1} + {a_n}]$
So, putting values in the above equation. We get,
So, putting values in the above equation. We get,
$\Rightarrow {S_{167}} = \dfrac{{167}}{2}[3 + 999] = \dfrac{{167}}{2}*1002 = 167*501 = 83667$
$\Rightarrow$Hence, the sum of all odd numbers between 1 and 1000 which are divisible by 3 is ${S_{167}} = 83667$.

Note: Whenever we came up with this type of problem then first, we find value of n using value of ${{\text{n}}^{th}}$ term formula in an A.P and then, we can easily find sum of n terms of that A.P using formula of sum of n terms of A.P, if first and last term are given.
Bookmark added to your notes.
View Notes
Sum of Odd Numbers  Sum of Odd Numbers Formula  Sequences and Series  Multiplication and Division of Integers  Even and Odd Numbers  Even and Odd Functions  Multiplication and Division of Integers - Rules  Properties of Integers  Operations of Integers  Integers  