Question

The sum of three consecutive odd natural numbers is 249. Find the numbers.

Answer 1

Here, we need to find the three consecutive odd natural numbers. Natural numbers are the integers greater than 0. Here, we will choose the three consecutive odd natural numbers $x$, $x + 2$, and $x + 4$. We will add these numbers and equate it with 249 to find the value of $x$. Then, we will substitute the value of $x$ in $x + 2$, and $x + 4$ to find all the numbers.

Complete step by step solution:
The numbers which come after each other are called consecutive numbers.
For example, 1 and 2 are consecutive numbers, 2 and 4 are consecutive even numbers (since no other even number comes between 2 and 4), and 3 and 5 are consecutive odd numbers.
Let the first odd natural number be $x$.
Thus, we get the next two odd natural numbers as $x + 2$ and $x + 4$.
Now, it is given that the sum of the three consecutive odd natural numbers is 249.
Thus, we get the equation in terms of $x$ as
$x + \left( {x + 2} \right) + \left( {x + 4} \right) = 249$
Adding the like terms on the left hand side of the equation, we get
$\Rightarrow 3x + 6 = 249$
Subtracting 6 from both sides, we get
$\begin{array}{l} \Rightarrow 3x + 6 - 6 = 249 - 6\\\ \Rightarrow 3x = 243\end{array}$
Finally, we will divide both sides by 3 to get the value of $x$.
$\begin{array}{l} \Rightarrow \dfrac{{3x}}{3} = \dfrac{{243}}{3}\\\ \Rightarrow x = 81\end{array}$
$\therefore$ We get the first number as 81.
We will find the next two odd natural numbers by substituting 2 in the expressions $x + 2$ and $x + 4$.
Second number $= x + 2 = 81 + 2 = 83$
Third number $= x + 4 = 81 + 4 = 85$

Therefore, the three consecutive odd natural numbers are 81, 83 and 85.

Note:
We should remember that the difference between any two consecutive odd numbers is 2. A common mistake is to let the numbers be $x$, $x + 1$, and $x + 2$. This gives the final answer as the numbers 82, 83, 84. This is not correct.
We can also verify our answer to make sure we did not make a calculation mistake. The sum of the three consecutive odd natural numbers is $81 + 83 + 85 = 249$. Therefore, our answer is correct.