What are the 4 concecutive even integers such that the squar of the second term is equal to the product of the first and last term
The even integers are:
a1 = 2 n
a2 = 2 ( n + 1 )
a3 = 2 ( n + 2 )
a4 = 2 ( n + 3 )
a2 ^ 2 = a1 * a4
[ 2 ( n + 1 ) ] ^ 2 = 2 n * 2 ( n + 3 )
4 ( n + 1 ) ^ 2 = 4 n * ( n + 3 ) Divide both sides by 4
( n + 1 ) ^ 2 = n * ( n + 3 )
n ^ 2 + 2 * n * 1 + 1 ^ 2 = n * n + n * 3
n ^ 2 + 2 n + 1 = n ^ 2 + 3 n Subtract n ^ 2 to both sides
n ^ 2 + 2 n + 1 - n ^ 2 = n ^ 2 + 3 n - n ^ 2
2 n + 1 = 3 n Subtract 2 n to both sides
2 n + 1 - 2 n = 3 n - 2 n
1 = n
n = 1
Solution:
a1 = 2 n = 2 * 1 = 2
a2 = 2 ( n + 1 ) = 2 ( 1 + 1 ) = 2 * 2 = 4
a3 = 2 ( n + 2 ) = 2 * ( 1 + 2 ) = 2 * 3 = 6
a4 = 2 ( n + 3 ) = 2 * ( 1 + 3 ) = 2 * 4 = 8
Proof :
a2 ^ 2 = a1 * a4
4 ^ 2 = 2 * 8
16 = 16
Remark :
a1 = 2 n
a2 = 2 ( n + 1 )
a3 = 2 ( n + 2 )
a4 = 2 ( n + 3 )
n is an integer.
suppose we just define n to be an even number
(that way I don't have to worry about the 2n idea)
then the 4 consecutive even number are
n , n+2, n+4, and n+6
(n+2)^2 = n(n+6)
n^2 + 4n + 4 = n^2 + 6n
4 = 2n
n = 2
so the 4 numbers are
2 , 4, 6, and 8
To find the four consecutive even integers, we can use algebraic expressions. Let's assume the first even integer as x, then the consecutive even integers will be x+2, x+4, and x+6 since the difference between consecutive even integers is always 2.
The problem states that the square of the second term (x+2)^2 is equal to the product of the first and last term, which is x * (x+6).
Now we can write the equation:
(x+2)^2 = x * (x+6)
To solve this equation, we will expand the squared term:
(x^2 + 4x + 4) = x^2 + 6x
Next, we will simplify the equation by combining like terms:
x^2 + 4x + 4 = x^2 + 6x
By subtracting x^2 from both sides of the equation:
4x + 4 = 6x
We can now isolate the variable x by moving the 4x term to the right side of the equation:
4 = 6x - 4x
Simplifying further:
4 = 2x
To solve for x, divide both sides by 2:
2 = x
So the first even integer is x = 2.
Plugging the value of x back into the consecutive even integer expressions:
The four consecutive even integers are: 2, 4, 6, 8.