Question

An even integer can be represented by the expression 2n, where n is any integer. Find three consecutive even integers that have a sum of 54. The three consecutive even integers, in order from least to greatest, are , , and . Explain your reasoning. Expressions for the next two consecutive even integers after 2n are and . Solving the equation =54, and substituting the solution into the expressions for the three integers results in the three integers given above.

Answer 1

As consecutive odd numbers differ by two (example: 3, 5, 7), the first odd number can be expressed as 2n + 1, the next can be found by adding two to the first to get 2n + 1 + 2 which simplifies to 2n + 3. Finally the expression for the third consecutive odd integer can be found by adding two to the previous, 2n + 3, to get 2n + 5. Adding these three together and setting them equal to your sum gets the equation

2n + 1 + 2n + 3 + 2n + 5 = 63

Combine like terms and solve For n.

Once you have n, you must substitute it back into your three expressions (2n + 1, 2n + 3, 2n + 5) to find the three odd integers.

Hope this helps :)

Answer 2

The numbers are 16, 18, and 20.

Even number

An even number is a number that can be divided into two equal parts. therefore, an even number can be given by '2n', as it is always divisible by two.

Given to us,

First even number = 2n

The sum of the three consecutive even numbers is 54.

As the first even number is 2n, the next two even numbers will be (2n+2), and (2n+4). We can not take the number as (2n+1) because it will always give an odd number. therefore,

2n + (2n+2) + (2n+4) = 54

2n + 2n + 2 + 2n + 4 = 54

6n + 6 = 54

Taking 6 as common,

6(n+1) = 54,

$\bold{n+1=\dfrac{54}{6}}$

n+1 = 9

n = 8.

first even number, 2n = 2x8 = 16,

second even number, (2n+2) = 2x8 + 2 = 18,

third even number, (2n+4) = 2x8 + 4 = 20,

Hence, the numbers are 16, 18, and 20.

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