Question

The average of all odd numbers up to $100$ is
A $49$
B $49.5$
C $50$
D $51$

Answer 1

As we know that the average of number is = $\dfrac{{{\text{Sum of all the term}}}}{{{\text{Total number of term }}}}$ when we write the odd term it form AP $1,3,5,7...............99$ so for find the number of term use formula of AP ${a_n} = a + (n - 1)d$ and for the summation of all the term is=$\dfrac{{{\text{Number of term }} \times {\text{ }}\left( {{\text{First term + Last Term }}} \right)}}{2}$ find the values and put it in the average equation .

Complete step-by-step answer:
As in the question we have to find the average of all odd numbers up to $100$ , if we write the odd term up to $100$ that is ,
$1,3,5,7...............99$
hence these term are in AP with
first term is $a = 1$
Common difference $d = 2$ and the last term is $l = 99$
let n be the total number of term present in this AP $1,3,5,7...............99$
As we know that the ${n^{th}}$ term or last term of this AP is $l = 99$
we know the formula that is ${a_n} = a + (n - 1)d$ where $a = 1$ , $d = 2$ , ${a_n} = 99$ we have to find n
On putting the given values
$\Rightarrow$ $99 = 1 + (n - 1)2$
$\Rightarrow$ $98 = (n - 1)2$
On dividing by $2$ in whole equation we get $49 = n - 1$
Hence $n = 50$
Total number of term in AP is $n = 50$

Sum of all odd number that is , $1 + 3 + 5 + 7 + ............. + 99$
we know the formula that
Sum of series = $\dfrac{{{\text{Number of term }} \times {\text{ }}\left( {{\text{First term + Last Term }}} \right)}}{2}$

first term is $a = 1$
the last term is $l = 99$
Total number of term in AP is $n = 50$
On putting it in the equation we get ,
$\Rightarrow$ $\dfrac{{50(1 + 99)}}{2}$
On solving we get Sum of all odd number is $2500$
For average of odd number is = $\dfrac{{{\text{Sum of all the term}}}}{{{\text{Total number of term }}}}$
$\Rightarrow$ $\dfrac{{2500}}{{50}}$ = $50$
Hence The average of all odd numbers up to $100$ is $50$ , or option C will be correct .

Note: As in the we use the summation of A.P terms = Number of term (First term + last term ) divided by $2$.We can also use instead of this is $\dfrac{n}{2}(2a + (n - 1)d)$ where n = number of term that is $50$ a= first term that is $1$ d = common difference that is $2$ of the A.P from here we will find the summation of AP .