SOLUTION: for any three consecutive numbers show that the difference between the product of the second and third and the product of the first and second is equal to double the second number.

Question 1008219: for any three consecutive numbers show that the difference between the product of the second and third and the product of the first and second is equal to double the second number.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let the first number be equal to x.
then the second number is equal to x+1
then the third number is equal to x+2

first number * second number = x * (x+1) = x^2 + x

second number * third number = (x+1) * (x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2

second number * third number minus first number * second number equals:

x^2 + 3x + 2 minus x^2 + x, which results in 2x + 2

2 * second number = 2 * (x+1) = 2x + 2.

they're the same.

i believe there is an assumption that the numbers are ascending in order and that the product of the first and second number is subtracted from the product of the second and third number.

if that's true, this will work whether the numbers are positive or negative.

for example:

2,3,4
product of first and second = 6
product of second and third = 12
12 - 6 = 6 which is equal to 2 * 3

-4,-3,-2
product of first and second = 12
product of second and third = 6
6 - 12 = -6 which is equal to 2 * -3

-1,0,1
product of first and second = 0
product of second and third = 0
0 - 0 = 0 which is equal to 2 * 0

0,1,2
product of first and second = 0
product of second and third = 2
2 - 0 = 2 which is equal to 2 * 1

-2,-1,0
product of first and second = 2
product of second and third = 0
0 - 2 = -2 which is equal to 2 * -1

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from purplemath:

Also note that order is important in the "quotient/ratio of" and "difference between/of" constructions. If a problems says "the ratio of x and y", it means "x divided by y", not "y divided by x". If the problem says "the difference of x and y", it means "x � y", not "y � x".
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that's why i changed your problem statement from:

for any three consecutive numbers show that the difference between the product of the first and second and the product of the second and third is equal to double the second number.

to:

for any three consecutive numbers show that the difference between the product of the second and third and the product of the first and second is equal to double the second number.

the purplemath reference i got this from can be found here:

http://www.purplemath.com/modules/translat.htm
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