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

Answer: (A)

Sum of odd numbers between 1 and 1000 , which is
divisible by $3=3+9+15+21+27+.......+999=S$ (let)
$\therefore$ Let $n$ be the number of terms in series and $a$ is first term.
$\therefore l=a+(n-1)d$,
where $l$ is last term and $d$ is is common difference.
$999=3+(n-1)\times 6$
$n-1=\frac{999-3}{6}=\frac{996}{6}$
$\Rightarrow n-1=166$
$\Rightarrow n=167$
$\therefore S=\frac{n}{2}[2a+(n-1)d]=\frac{167}{2}[2\times 3+(167-1)\times 6]$
$=\frac{167}{2}[1002]=167\times 501=83667$