Question

What is the value of $1+2+3+4+.....+100$?

Answer 1

Hint: We first find the general formula of summation of first n natural numbers as ${{S}_{n}}=\dfrac{n\left( n+1 \right)}{2}$. We replace the value with $n=100$ to find the multiplication. We complete the division to find the final solution.

Complete step-by-step answer:
We have given a series of $1+2+3+4+.....+100$.
This is the sum of the first 100 natural numbers.
We first find the general form of such sum.
If we need the sum of first n natural numbers then it can be expressed with the formula of
${{S}_{n}}=\dfrac{n\left( n+1 \right)}{2}$.
Now we can place the value of 100 in the place of n as $n=100$ to get the value of ${{S}_{100}}$.
Therefore, ${{S}_{100}}=1+2+3+4+.....+100$.
We have ${{S}_{100}}=1+2+3+4+.....+100={{S}_{n}}=\dfrac{100\left( 100+1 \right)}{2}=\dfrac{100\times 101}{2}$.
We can see that 2 will divide the number 100.
For any fraction $\dfrac{p}{q}$, we first find the G.C.D of the denominator and the numerator. If it’s 1 then it’s already in its simplified form and if the G.C.D of the denominator and the numerator is any other number d then we need to divide the denominator and the numerator with d and get the simplified fraction form as $\dfrac{{}^{p}/{}_{d}}{{}^{q}/{}_{d}}$.
For our give fraction $\dfrac{100}{2}$, the G.C.D of the denominator and the numerator is 2.
$\begin{align}
  & 2\left| \!{\underline {\,
  2,100 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,50 \,}} \right. \\
\end{align}$
Now we divide both the denominator and the numerator with 2 and get
$\dfrac{100\times 101}{2}=\dfrac{100}{2}\times 101=50\times 101=5050$.
Therefore, the value of $1+2+3+4+.....+100$ is 5050.

Note: In case of the starting number is m for the summation of n numbers then we can also find the summation in the form of ${{S}_{n+m-1}}-{{S}_{m-1}}$. We just add the previous numbers starting from 1 to find the similar form of ${{S}_{n}}=\dfrac{n\left( n+1 \right)}{2}$ and then subtract the extra numbers.