Question

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

Answer 1

Hint: The first odd number between $0$ and $50$ is $1$ and the last odd number between $0$ and $50$ is $49$.The odd numbers between $0$ and $50$ are in arithmetic progression as they have a common difference of $2$. Here, we can find the total number of terms by the formula-
$ \Rightarrow {T_n} = a + \left( {n - 1} \right)d$ Where ${T_n}$ is the last term of AP, ‘a’ is the first term of the series, n is number of terms and d is the common difference. Now, to solve this question we can use the formula of the sum of AP which is given as-
$ \Rightarrow {S_n} = \dfrac{n}{2}\left\{ {2a + \left( {n - 1} \right)d} \right\}$ Where ‘a’ is the first term of the series, n is number of terms and d is the common difference. Put the values in this formula and solve.

Complete step-by-step answer:
We have to find the sum of odd numbers between $0$ and $50$.We know that the first odd number between $0$ and $50$ is $1$ and the last odd number between $0$ and $50$ is $49$.
Then the series of odd numbers can be written as- $1,3,5......,47,49$
Here we see that the difference between the consecutive terms is $2$. So the series is in Arithmetic progression as it has a common difference of $2$.
So first we have to find the total number of terms in this series. We can use the formula of the last term of AP to find the total number of terms which is given by-
$ \Rightarrow {T_n} = a + \left( {n - 1} \right)d$ Where ${T_n}$ is the last term of AP, ‘a’ is the first term of the series, n is number of terms and d is the common difference.
So putting the given values, we get-
$ \Rightarrow 49 = 1 + \left( {n - 1} \right)2$
On solving, we get-
$ \Rightarrow 49 = 1 + 2n - 2$
On simplifying we get-
$ \Rightarrow 49 = 2n - 1$
$ \Rightarrow 2n = 49 + 1 = 50$
Then on further solving, we get-
$ \Rightarrow n = \dfrac{{50}}{2} = 25$
Now we know the number of terms so we can use the formula of sum of n terms of AP which is given as-
$ \Rightarrow {S_n} = \dfrac{n}{2}\left\{ {2a + \left( {n - 1} \right)d} \right\}$ Where ‘a’ is the first term of the series, n is number of terms and d is the common difference
On putting the given values in the formula, we get-
$ \Rightarrow {S_n} = \dfrac{{25}}{2}\left\{ {2 + \left( {25 - 1} \right)2} \right\}$
On simplifying we get-
\[ \Rightarrow {S_n} = \dfrac{{25}}{2}\left\{ {2 + 24 \times 2} \right\} = \dfrac{{25}}{2}\left\{ {2 + 48} \right\}\]
On simplifying further, we get-
\[ \Rightarrow {S_n} = \dfrac{{25}}{2} \times 50\]
On solving, we get-
\[ \Rightarrow {S_n} = 25 \times 25 = 625\]
The sum of odd numbers between $0$ and $50$ is $625$.

Note: The students may get confused by the dilemma of whether they should count zero as an odd number or not. So remember that zero is neither odd nor even. It is a unique number that shows unique properties.-
It is a whole number.
When it is added with an odd number it will give an odd number and when it is added with an even number it will give an even number.
It lies between two odd numbers $1$ and $ - 1$ in the number line so it is considered an even number by many people.
So if the students count zero as an odd number then the number of terms will change. This will give an incorrect answer.