Question

Find the sum of first 100natural numbers​

Answer 1

Given ,

The AP is 1,2,3, .... ,100

Here ,

First term (a) = 1

Common difference (d) = 1

Last term (l) = 100

We know that , the sum of first n terms of an AP is given by

$\star \: \: \sf S_{n} = \frac{n}{2} (a + l)$

Thus ,

$\sf \Rightarrow S_{100} = \frac{100}{2} (1 + 100) \\ \\ \sf \Rightarrow S_{100} =50 \times (101) \\ \\ \sf \Rightarrow S_{100} =5050$

$\therefore \bold{ \sf \underline{The \: sum \: of \: first \: 100 \: terms \: of \: natural \: no. \: is \: 5050 \:}}$

Answer 2

Question :-

Find the sum of first 100 natural numbers

Answer :-

Required to find :-

Sum of first 100 natural numbers ?

Concept used :-

Arithmetic progession

Formula used :-

$\large{\tt{ {S}_{nth} = \dfrac{n}{2} \times ( 2a + ( n - 1 ) d ) }}$

or

$\large{\tt{ {S}_{nth} = \dfrac{n}{2} \times [ first \; term + last \; term ]}}$

Here, n refer to the no. of terms

Try using the 1st formula because it is useful in most of the questions

Solution :-

Natural numbers :-

1,2,3,4, - - - - - 100

This can be written as an Arithmetic progession .

Because,

The difference between the numbers is constant .

That is

( 2nd term - 1st term ) = ( 3rd term - 2nd term) = ( 4th term - 5th term )

1 = 1 = 1

Since, the above one satisfies this condition .

So, this can be written in the form of ,

$\tt{ {a}_{nth} = {a}_{1} , {a}_{2} , {a}_{3} , {a}_{4} - - - - , {a}_{nth}}$

Hence,

$\tt{ {a}_{100} = 1 , 2, 3 ,4 , - - - - , 100 }$

Here,

The first term is 1 .

Similarly, First term is represented by a

So, a = 1

And

The common difference = 1

Similarly, common difference is denoted by d .

So, d = 1

Now using the formula ,

$\large{\tt{ {S}_{nth} = \dfrac{n}{2} \times ( 2a + ( n - 1 ) d ) }}$

Here,

n is the number till which we need to find the sum .

So,

Here,

$\longrightarrow{\tt{ {S}_{nth} = {S}_{100} }}$

Hence,

The required values are,

• n = 100
• a = 1
• b = 1

$\tt{ {S}_{100} = \dfrac{100}{2} \times ( 2(1) + (100 - 1)1 )}$

$\tt{ {S}_{100} = 50 \times ( 2 + 99 )}$

$\tt{ {S}_{100} = 50 \times 101 }$

$\red{\underline{\underline{ {S}_{100} = 5050 }}}$

So,

The sum of first 100 natural numbers is 5050 .

Points to remember :-

a = first term

d = common difference

$\large{\tt{ {S}_{nth} = \dfrac{n}{2} \times ( 2a + ( n - 1 ) d ) }}$

$\large{\tt{ {a}_{nth} = a + ( n - 1 ) d }}$

This formula is useful for finding the nth term of a sequence .