Question

Show that the sum of all odd integers between 1 and 1000
which are divisible by 3 is 83667.

Answer 1

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, $$a_n=999,a_1=3$$ and $$d=6$$ in the above equation. We get,
$$\begin{gathered} \Rightarrow 999=3+(n-1)6 \\ \Rightarrow 999=3+6n-6 \end{gathered}$$
On solving the above equation. We get,
$$\Rightarrow n={\displaystyle \frac{1002}{6}}=167$$ numbers in the A.P
Now, as we know that sum of these n terms of A.P is given by,
$$\Rightarrow S_n={\displaystyle \frac{n}{2}}[a_1+a_n]$$
So, putting values in the above equation. We get,
So, putting values in the above equation. We get,
$$\Rightarrow S_{167}={\displaystyle \frac{167}{2}}[3+999]={\displaystyle \frac{167}{2}}\ast 1002=167\ast 501=83667$$
$$\Rightarrow$$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.