Question

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

Answer 1

lAll odd numbers between $1$ and $1000$ which are divisible by $3$, are

$3,9,15,......999$

which forms an A.P

first term of this A.P is $a_1=3$

second term of this A.P is $a_2=9$

last term of this A.P is $a_n=999$

common difference

$d=a_2-a_1$

$⟹d=9-3=6$ .

nth term of this A.P is given by

$a_n=a_1+(n-1)d$

put $a_n=999;a_1=3andd=6$ in above equation we get,

$⟹999=3+(n-1)6$

$⟹6n-6+3=999$

$⟹6n=999+3=1002$

$n=\frac{1002}{6}=167$ number of terms in this A.P

now, sum of these n=167 terms is given by

$S_n=\frac{n}{2}(a_1+a_n)$

put values of $n=167;a_1=3;a_n=999$ we get

$S_{167}=\frac{167}{2}(3+999)$

$⟹S_{167}=\frac{167}{2}\times 1002$

$⟹S_{167}=167\times 501$

$⟹S_{167}=83667$

hence the sum of all odd numbers between 1 and 1000 which are divisible by $3$, is $S_{167}=83667$