Well what i am talking about is you know how they have for example: Step 2, and they have 6 arrows in square shape! And as you go up the steps the arrows increase in size, forming a number of different boxes. (like how many arrows would be needed for step 5) the question is: Write a rule in your own words to explain how you could calculate, from the step number, the number of arrows that would be used in this step.
okay, so here is what we have:
steps...........arrows
..1...................5
..2...................6
..3...................7
..4...................8
..5...................?
do you see a pattern with the number of arrows? sure, we're just going up by one correct?
so, step 5 -----> 9 arrows
step 6 --------->10 arrows
step 7 --------->11 arrows
and so on...
okay. let x be the current step we are at, and let f(x) be the function that gives the amount of arrows needed for that step.
then the pattern is given by the formula:
f(x) = x + 4
so whatever step number we want, we just replace x with that value, and we have it. test it out
we know at the 3rd step we had 7 arrows.
f(3) = 3 + 4 = 7 eureka!
how many arrows are at the 56th step? well, there are f(56) arrows of course.
f(56) = 56 + 4 = 60 arrows!
do you know how i came up with the formula?
Here's how i came up with it.
what we want here is called a function. basically it is a relationship that relates a pair of numbers. i wanted a relationship that links the number of step i am to the number of arrows needed for that step. so my formula must depend on the current step i am working with. so what i did was try to rewrite all the numbers on the right using the step that they were at. and i had to make sure that this formula worked for all of them. after working for a few, we can be pretty much certain it will work for the rest. here goes:
steps...........arrows
..1...................5 = 1 + 4
..2...................6 = 2 + 4
..3...................7 = 3 + 4
..4...................8 = 4 + 4
..5...................9 = 5 + 4
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..x...................f(x) = x + 4
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so x represents an arbitrary step. i realized that for whatever step i was on, i just needed to add 4 to it to get the number of arrows, so if i'm on the x step (whatever x is) i just need to add 4 to it to get the number of arrows and so my formula for the arrows is x + 4 where x is the current step
f(x) just means a "function of x" which as i said, is just a relationship between x and another number (namely the number of arrows)