Question

What are four consecutive even integers whose sum is 108?

Answer 1

Hint: We first try to find the general form of even integers. The form of the integers will be $ 2k,k\in \mathbb{Z} $ . We know that the difference between any two consecutive even integers is always equal to 2. We use that relation to find the integers. We find the additional value of the integers. This gives the value of 108. We solve to find the value of $ n $ .

Complete step by step solution:
We have been given that the sum of three consecutive even integers is 108.
We can define any even integers in the form of $ 2k,k\in \mathbb{Z} $ .
The difference between any two consecutive even integers is always equal to 2.
Now we assume those three consecutive even integers where the integers are $ 2n-2,2n,2n+2,2n+4 $ .
Now if we add these integers the answer would be 108. We express this relation in a mathematical form and get $ \left( 2n-2 \right)+2n+\left( 2n+2 \right)+\left( 2n+4 \right)=108 $ .
We now solve the equation to find the solution for the variable $ n $ .
We separate the variables and the constants of the equation $ \left( 2n-2 \right)+2n+\left( 2n+2 \right)+\left( 2n+4 \right)=108 $ .
 $ \begin{align}
  & \left( 2n-2 \right)+2n+\left( 2n+2 \right)+\left( 2n+4 \right)=108 \\
 & \Rightarrow 2n-2+2n+2n+2+2n+4=108 \\
 & \Rightarrow 8n=104 \\
\end{align} $
Now we divide the both sides of the equation $ 8n=104 $ with 8 to get
 $ \begin{align}
  & \dfrac{8n}{8}=\dfrac{104}{8} \\
 & \Rightarrow n=13 \\
\end{align} $
The value of the variable $ n $ is $ n=13 $ .
We need to find the values of the integers $ 2n-2,2n,2n+2,2n+4 $ .
Multiplying the equation $ n=9 $ with 2, we get
 $ \begin{align}
  & n\times 2=13\times 2 \\
 & \Rightarrow 2n=26 \\
\end{align} $
Therefore, the integers are $ 2n-2=24,2n=26,2n+2=28,2n+4=30 $ .
So, the correct answer is “24,26,28,30”.

Note: We assumed the values where we first assumed the middle number and then found the other three. This helps in eliminating the constant term from the addition. We also could have used the variable $ n $ directly for the odd number between the two middle even numbers. The set of numbers would have been $ n-3,n-1,n+1,n+3 $ .