Question

Find the sum of all odd numbers between 0 and 50.

Answer 1

HEY THERE!!!

Let to be First term 'a' and Common difference 'd'.

Firstly,all odd numbers between 0 and 50 which are given below in the form of Arithmetic Sequence or Progression;

1,3,5,7,9,11.....49

Now, Write in Summation of Arithmetic Sequence or Progression;-

Let Sn = 1+3+5.....49

Here,

First term (a)= 1

Common difference (d)=2

Last terms= 49

Let the number of terms be 'n'

Then,

Tn =a+(n-1)d

Substitute the value of the Required terms in Above Formula;-

Tn =a+(n-1)d

=> 1+(n-1)2 = 49

=> 1+2n-2= 49

=> -1+2n=49

=>2n=49+1

=> 2n=50

=> n=50/2

=> n=25

Now, Using Formula of Summation of terms which known as Sum of terms;-

$\implies \: Sn \: = \frac{n}{2} (a + l) \\ \\ \\ \implies \: Sn \: = \frac{25}{2} (1+ 49) \\ \\ \implies \: Sn = \frac{25}{ 2} {(50)}\\ \\ \\ \implies \: Sn = \frac{25}{ \cancel{2} }{( \cancel50)} \\ \\ \\ \\ \implies \: 25 \times 25 = 625 \\ \\ \\ \\ \implies \: 625$

Hence, 625 are the sum of all odd numbers between 0 and 50.

Answer 2

$\underline{\underline{\mathfrak{\large{Solution : }}}}$




$\underline{\mathsf{To \: Find \: \longrightarrow Sum \: of \: all \: odd \: no.s \: }} \\ \underline { \mathsf{between \: 0 \: and \: 50.}}$




$\textsf{We know that : } \\ \\ \mathsf{\longrightarrow First \: odd \: no. \: after \: 0 \: is \: 1 .} \\ \\ \mathsf{\longrightarrow Last \: odd \: no. \: before \: 50 \: is \: 49.} \\ \\ \mathsf{\longrightarrow Difference \: between \: two \: consecutive \: odd \: no.s \: are \: 2.}$




$\textsf{The mathematical form for our solution is : }$



$\\ \mathsf{= \: 1 \: + \: 3 \: + \: ............. \: + \: 49} \\ \\ \textsf{The above series forms an Arithmetic Progression.} \\ \\ \textsf{Where,} \\ \\ \mathsf{\longrightarrow First \: term (a) \: = \: 1 } \\ \\ \mathsf{\longrightarrow Common \: difference (d) \: = \: 2 } \\ \\ \mathsf{\longrightarrow Last \: term (l) \: = \: 49}$


$\underline{\textsf{Using Formula : }} \\ \\ \boxed{\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}(a \: + \: l)}}$




$\textsf{So, in order to find sum of this series.} \\ \textsf{First of all, we need to find its no. of terms.}$



$\underline{\textsf{Now,}} \\ \\ \underline{\textsf{Using Formula : }} \\ \\ \boxed{\mathsf{\implies l \: = \: a \: + \: (n \: - \: 1))d}}$



$\mathsf{\implies 49 \: = \: 1 \: + \: ( n \: - \: 1)2} \\ \\ \mathsf{\implies 49 \: - \: 1 \: = \: ( n \: - \: 1)2} \\ \\ \mathsf{\implies 48 \: = \: (n \: - \: 1)2} \\ \\ \mathsf{\implies \dfrac{48}{2} \: = \: ( n \: - \: 1)} \\ \\ \mathsf{\implies 24 \: = \: n \: - \: 1} \\ \\ \mathsf{\implies n \: = \: 24 \: + \: 1 } \\ \\ \mathsf{\therefore \quad n \: = \: 25}$




$\underline{\large{\textsf{Now,}}}$


$\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}( a \: + \: l)} \\ \\ \mathsf{\implies S_{25} \: = \: \dfrac{25}{2}( 1 \: + \: 49 )} \\ \\ \mathsf{\implies S_{25} \: = \: \dfrac{25}{2} \: \times \: 50} \\ \\ \mathsf{\implies S_{25} \: = \: 25 \: \times \: 25} \\ \\ \mathsf{\therefore \quad S_{25} \: = \: 625}$





$\boxed{\large{\textsf{The required sum is 625.}}}$







$\underline{ \textsf{More to know : }} \\ \\ \mathsf{\implies Sum \: of \: first \: n \: odd \: no.s \: = \: {n}^{2}}$