Question

What is the sum of the first 100 natural numbers?

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 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 given 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.
So, the correct answer 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.