Question

What is the sum of all natural numbers from 1 to 100?
$
  (a){\text{ 5050}} \\
  (b){\text{ 50}} \\
  (c){\text{ 4550}} \\
  (d){\text{ 5150}} \\
 $

Answer 1

Hint: In this question we have to find the sum of all natural numbers from 1 to 100. Now try and form a series of all natural numbers between the given ranges. On careful observation the common difference that is the difference between the consecutive terms will be found to be constant, thus it forms an A.P. Use the basic A.P series formula for sum to get the answer.

Complete step-by-step answer:
As we know (1, 2, 3, 4………………………, 100) forms an A.P with first term (a = 1), common difference (d = (2 -1) = (3 - 2) =1) and the number of terms (n =100).
Now we know the formula of sum of an A.P which is given as
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ Where ( a is first term, d is common difference and n is number of terms).
So the sum of all natural numbers from 1 to 100 is
$ \Rightarrow {S_{100}} = \dfrac{{100}}{2}\left( {2\left( 1 \right) + \left( {100 - 1} \right)1} \right)$
Now simplify the above equation we have,
$ \Rightarrow {S_{100}} = 50\left( {2 + 99} \right) = 50\left( {101} \right) = 5050$
So this is the required sum of all natural numbers from 1 to 100.
So this is the required answer.
Hence option (A) is correct.

Note: Whenever we face such types of problems the key concept is to figure out that whether the series which is forming makes up an A.P, G.P or H.P. then depending upon the observations use the respective series formula. This concept along with basic understanding of the series will help getting on the right track to reach the answer.