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Q.
The sum of all odd numbers between $1$ and $1000$ which are divisible by $3$ is
Sequences and Series
Solution:
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 $
$\Rightarrow n=167$
$\therefore S=\frac{n}{2}[2 a+(n-1) d]=\frac{167}{2}[2 \times 3+(167-1) \times 6]$
$=\frac{167}{2}[1002]=167 \times 501=83667$