Question

What are the three consecutive even integers whose sum is $ 54 $ ?

Answer 1

Hint: In the given question, we are required to find the three consecutive even integers whose sum totals up to $ 54 $ . The given question involves the concepts of series, sequences and progressions. We know that consecutive even integers are in arithmetic progression as the difference between the consecutive terms is constant. We have to find the terms in arithmetic progression whose sum is given to us as $ 54 $ .

Complete step-by-step answer:
So, the three consecutive even integers are in arithmetic progression.
Also, we know that every even integer is of form $ 2n $ and the difference between two consecutive even integers is $ 2 $ .
So, we suppose that the consecutive even integers are of the form: $ \left( {2n - 2} \right) $ , $ \left( {2n} \right) $ , $ \left( {2n + 2} \right) $ .
So, we are given that the sum of these numbers is $ 54 $ .
Hence, we get, $ \left( {2n - 2} \right) + \left( {2n} \right) + \left( {2n + 2} \right) = 54 $
Cancelling the equivalent terms with opposite signs and adding up the rest of the terms, we get,
 $ \Rightarrow 6n = 54 $
Now, dividing both sides of the equation by $ 6 $ , we get,
 $ \Rightarrow n = 9 $
So, the value of n is $ 9 $ .
Hence, the consecutive even integers whose sum is $ 54 $ are: $ 2\left( 9 \right) - 2 $ , $ 2\left( 9 \right) $ , $ 2\left( 9 \right) + 2 $ .
 $ \Rightarrow 16,18,20 $
Therefore, the consecutive even integers whose sum is given to us as $ 54 $ are: $ 16,18,20 $
So, the correct answer is “ $ 16,18,20 $ ”.

Note: In such questions, we have to first assume the terms such that its easy for us to perform the operations and use the information provided to us in the question. The, we find the value of the assumed variable and hence get our required answer.