Question

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

Answer 1

Hint:
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.