Question

The sum of all odd numbers between $1$ and $1000$ which are divisible by $3$, is

• A. $83667$
• B. $90000$
• C. $83660$
• D. None of these

Answer 1

The correct option is A

$83667$

Explanation for correct option

Here the first term $a_1=3$

Second term $a_2=9$

Therefore common difference $d=6$

Let the total number of terms $n$

Last term $a_n=999$

We know that last term $a_n=a_1+d\left(n-1\right)$

$$\begin{gathered} \Rightarrow a_1+d\left(n-1\right)=999 \\ \Rightarrow 3+6\left(n-1\right)=999 \\ \Rightarrow n-1=\frac{999-3}{6} \\ \Rightarrow n=167 \end{gathered}$$

We know that Sum of $n$ terms$=\frac{n}{2}\left(a_1+a_n\right)$

$$\begin{gathered} =\frac{167}{2}\left(3+999\right) \\ =\frac{167}{2}\left(1002\right) \\ =167\times 501 \\ =83667 \end{gathered}$$

Hence, option A is correct.