POW #24, due Monday 5/4
Any three consecutive integers have an interesting property.
- Write down any three consecutive integers, such as 5, 6, 7.
- Square the second number.
- Multiply the first number by the third.
- Note how the numbers that you got in steps b and c are related.
Try this with at least three different sets of starting numbers.
- Show the integers you used each time and the calculations that you did, then compare the resulting two numbers.
- By using algebraic expressions with a variable, show that your conclusion will always be true.
Hint #1
To create an algebraic expression, designate one of your three consecutive integers as a variable, such as x, and then write expressions for your other two integers in terms of x.
Perform the same operations that you did with the exact integers with these variable expressions. The simplified result will be your “proof.”
Hint #2
Your choice of which consecutive integer to designate as x will have a big effect on the amount of algebra-arithmetic work you end up doing. In this case the best choice is probably the middle number.
By designating the middle number as x, one half of your comparison is simply x-squared. (I’ll leave the other half to you.)