Question

How do you add all the odd numbers between $1-99$ inclusive?

Answer 1

Finding the sum of odd numbers $1-99$:

Now the series becomes

$1+3+5+7+.....+99$

So we can see that this is an arithmetic series with common difference, $d=2$, first term, $a=1$and last term, $l=99$

We know that $l=a+(n-1)d$

$$\begin{gathered} \therefore 99=1+(n-1)\times 2 \\ \Rightarrow \left(n-1\right)=\frac{98}{2} \\ \Rightarrow n=89+1 \\ \Rightarrow n=50 \end{gathered}$$

We know that sum of $n^{th}$ term of $\mathrm{AP}.$

$$\begin{gathered} \Rightarrow s=\frac{n}{2}\left(a+l\right) \\ \therefore s=\frac{50}{2}\left(1+99\right) \\ \Rightarrow s=25\times 100 \\ \Rightarrow s=2500 \end{gathered}$$

Hence, the sum of all add numbers between $1-99$ $=2500$.