Question

Find the sum of all natural numbers from$1$ to $100.$

Answer 1

Solve for the sum of the first hundred natural numbers

The natural numbers from$1$ to$100$ can be written as

$1,2,3,4,5\dots \dots .100$ is an arithmetic progression (A.P), where first term $=a$, common difference $=d$, and number of terms $=n$ i.e

$$\begin{gathered} a=1 \\ d=1 \\ n=100 \end{gathered}$$

Let us calculate the sum of the natural numbers

The formula of the sum of an A.P series

$$\begin{gathered} \Rightarrow \mathit{S}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{[}\mathbf{2}\mathit{a}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}\mathit{n}\mathbf{}\mathbf{-}\mathbf{}\mathbf{1}\mathbf{)}\mathbf{}\mathbf{\times }\mathbf{}\mathit{d}\mathbf{]}\mathbf{}\mathbf{} \\ \Rightarrow S=\frac{100}{2}[2+(100–1)\times 1] \\ \Rightarrow S=50[2+99] \\ \Rightarrow S=5050 \end{gathered}$$

Hence, the sum of all natural numbers from $1$to $100$ is $5050.$