Question

Find three consecutive odd natural numbers whose sum is $87$ .

Answer 1

Hint: We all know that the numbers which are not divisible by $2$ are called odd numbers. For example, $1,7,15,71,93,...$ . As we also know that the consecutive numbers are the numbers which are in series or continuation. For example, $\left( {1,2,3} \right),\left( {10,11,12} \right)$ . So, the consecutive odd numbers can be represented as a difference of $2$ from a variable $x$ . For example, $x,\left( {x + 2} \right),\left( {x + 4} \right)$ which can be $\left( {11,13,15} \right),\left( {21,23,25} \right)$ . Their sum is given in the question and we can equate them and find the three consecutive odd numbers.

Complete step-by-step answer:
Let us assume an odd number$x$ which is not divisible by $2$ .
The three consecutive odd numbers can be written as $x,x + 2,x + 4$ .
The sum of these numbers can be represented as $x + \left( {x + 2} \right) + \left( {x + 4} \right)$
According to the given question, the sum of three consecutive odd numbers is $87$ .
Now, equate both the sums:
 $
\Rightarrow x + \left( {x + 2} \right) + \left( {x + 4} \right) = 87 \\
\Rightarrow 3x + 6 = 87 \\
\Rightarrow 3x = 81 \\
\Rightarrow x = \dfrac{{81}}{3} \\
\Rightarrow x = 27 \\
 $
The value of $x$ is $27$ .
Therefore, the three consecutive odd numbers are:
First number $ = x = 27$
Second number $ = x + 2 = 27 + 2 = 29$
Third number $ = x + 4 = 27 + 4 = 31$
Hence, three consecutive odd numbers whose sum is $87$ are $27,29{\text{ and }}31$ .

Note: The consecutive numbers should be represented as a difference of $1$ such as $x,\left( {x + 1} \right),\left( {x + 2} \right)$ but we have to careful with the words as given in question. The given question demands the consecutive odd numbers which will be represented as a difference of $2$ . We know that the general representation of an odd number is $\left( {2n + 1} \right)$ so we can continue with this actual representation also. This can be solved by taking $\left( {2n + 1} \right)$ and its consecutive odd numbers which can be written as:
 $\left( {2n + 1} \right),\left( {2n + 1 + 2} \right),\left( {2n + 1 + 4} \right)$
The simplified numbers become $\left( {2n + 1} \right),\left( {2n + 3} \right),\left( {2n + 5} \right)$
Now, we will equate its sum equal to $87$ .
 $
\Rightarrow 2n + 1 + 2n + 3 + 2n + 5 = 87 \\
\Rightarrow 6n + 9 = 87 \\
\Rightarrow 6n = 78 \\
\Rightarrow n = 13 \\
 $
Therefore, the three consecutive odd numbers will be:
First number $ = 2n + 1 = 2\left( {13} \right) + 1 = 27$
Second number $ = 2n + 3 = 2\left( {13} \right) + 3 = 29$
Third number $ = 2n + 5 = 2\left( {13} \right) + 5 = 31$