Question

Find the sum of odd integers from $1$ to $2001$.

Answer 1

Step $1:$ Finding first term, commom difference and last term.

Odd integers from $1$ to $2001$ are
$1,3,5,7,...,2001$
Clearly, this sequence is an A.P.
Where $a=1,d=2,l=2001$

Step $2:$ Finding number of terms
As we know,
$l=a+(n-1)d$
$\Rightarrow 2001=1+(n-1)\times 2$
$\Rightarrow (n-1)\times 2=2001-1$
$\Rightarrow (n-1)=\frac{2000}{2}=1000$
$\therefore n=1001$

Step $3:$ Finding sum of required odd numbers.

Let $S=1+3+5+\cdots +2001$
As we know, formula of sum of $n$ terms of an A.P.
$S_n=\frac{n}{2}[2a+(n-1\left)d\right]$
$=\frac{1001}{2}[2\times 1+(1001-1)\times 2]$
$=\frac{1001}{2}[2+2\times 1000]$
$=\frac{1001\times 2002}{2}=(1001{)}^2$
$=1002001$

Final answer: Hence, the sum of odd integers from $1$ to $2001$ is $1002001$.