Question

If the sum of three consecutive even integers is \[66\] what are the integers?

Answer 1

Hint: Here we are asked to find three consecutive even integers whose sum is given as sixty-six. In mathematics, the consecutive terms are generally expressed as $n,n + 1,n + 2,n + 3,n + 4,...$ and the even number is generally written as $2n$ then by using the general form of consecutive numbers we get the consecutive even numbers as $2n,2n + 2,2n + 4,2n + 6,...$ where $n \in \mathbb{Z}$ . Using this general form, we will find those three even integers.

Complete step-by-step answer:
The format of the even number is \[2n\] , \[n \in \mathbb{Z}\] .
Assume that the first even number is \[2n\] .
And the second even number is found to add on 2 in the first even number.
We find the next even number if we add $2$ .
So the second even number is \[2n + 2\] .
Then the third even number is \[2n + 4.\]
Let these three even numbers be $a_1$,$a_2$ and $a_3$.
${a_1} = 2n $
${a_2} = 2n + 2$
${a_3} = 2n + 4 $
Given, the Sum of these three numbers is \[66.\]
Which means
$a_1+a_2+a_3=66$ - (1)
\[\Rightarrow 2n + 2n + 2 + 2n + 4 = 66\]
Add to right-hand side terms,
\[6n + 6 = 66\]
Now subtract \[6\] on both sides,
\[6n + 6 - 6 = 66 - 6\]
Then get
\[6n = 60\]
Now divide \[6\] into both sides
\[n = 10\]
Now we get the \[n\] value is \[10\] .
We apply the \[n\] values in the first term \[{a_1}\]
\[n = 10\]
${a_1} = 2n $
$= 2(10) $
$= 20 $
The first term is \[20\] .
And applying the \[n\] value is the second term \[{a_2}\] .
$n = 10 $
${a_2} = 2n + 2 $
$= 2(10) + 2 $
$= 20 + 2 $
$= 22 $
The second term is \[22\] .
Then apply the \[n\] value in the third term \[{a_3}\] .
$n = 10 $
${a_3} = 2n + 4 $
$= 2(10) + 4 $
$= 20 + 4 $
$= 24 $
The third term is \[24.\]
Therefore, the three consecutive even numbers are \[20,22\,\] and \[24\] in respectively.

Note:
We can cross check whether our answer is correct or not by substituting the values in equation (1).
${a_1} + {a_2} + {a_3} = 66 $
$\Rightarrow 20 + 22 + 24 =66 $
$66 = 66 $
Which is true. Our values satisfy the equation which means our answer is correct.

> In this problem, we have formed an equation by using the given statement to find the value of $n$ which will be used to find the individual term easily.