Question

Find the sum of all the odd natural numbers less than 50 using the concept of Arithmetic progression (A.P).

Answer 1

- Hint: Here, we need to make the list of all odd natural numbers less than 50 i.e. 50 is not included in calculation. Further, we will be using summation formula of Arithmetic progression which is ${{S}_{n}}=\dfrac{n}{2}\left( a+l \right)$ where l is the last term of the series, a is first term and n is the total numbers in the series.

Complete step-by-step solution -

Now, we have to find odd numbers which can be easily done by dividing the numbers by 2 and which leave remainder not equal to zero are all odd numbers.
So, we get our series as 1,3,5,7,9, …… 47,49 which is in A.P having the same common difference between any two consecutive numbers.
Here, the first term $a=1,l=49,n=25$ which can be calculated by imagining that 1 to 50 is a total 50numbers but here we have alternate numbers, so total numbers are half of 50 i.e. n is 25.
Now, using summation formula:
${{S}_{n}}=\dfrac{n}{2}\left( a+l \right)$
Substituting all the value sin the formula, we get
${{S}_{25}}=\dfrac{25}{2}\left( 1+49 \right)$
${{S}_{25}}=\dfrac{25}{2}\left( 50 \right)$
${{S}_{25}}=25\times 25$
${{S}_{25}}=625$
Thus, the sum of odd natural numbers less than 50 is 625.

Note: Be careful while reading what is asked in question. Sometimes, chances of mistakes are not reading odd natural numbers less than 50 and simply calculating including 50 which results in the wrong answer. Also, be careful while using summation formula instead of using ${{n}^{th}}$ term finding formula which is \[{{T}_{n}}=a+\left( n-1 \right)d\] where is the common difference between two consecutive numbers.