Question

Find the sum of all odd numbers between $100$ and $200$

Answer 1

All odd numbers between $100$ and $200$ are

$101,103,105,......199$

which forms an $A.P$

first term of this A.P is $a_1=101$

second term of this A.P is $a_2=103$

last term of this A.P is $a_n=199$

common difference

$d=a_2-a_1$

$⟹d=103-101=2$ .

$n^{th}$ term of this $A.P$ is given by

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

put $a_n=199;a_1=101$and $d=2$ in above equation we get,

$⟹199=101+(n-1)2$

$⟹2n-2+101=199$

$⟹2n=199-99=100$

$n=\frac{100}{2}=50$ number of terms in this A.P

now, sum of these n=50 terms is given by

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

put values of $n=50;a_1=101;a_n=199$ we get

$S_{50}=\frac{50}{2}(101+199)$

$⟹S_{50}=\frac{50}{2}\times 300$

$⟹S_{50}=25\times 300$

$⟹S_{50}=7500$

hence the sum of all odd numbers between 100 and 200, is $S_{50}=7500$