Sorry if I sound a bit dim but I am in a bit of a panic as I have a son worried about his homework which is due in tomorrow. He has a sheet with two rows and five columns as follows:
Position 1 2 3 4
Term 9 13 17 21
The first question is:
a. What is the term to term rule? (how do you get from 9 to 13, 13 to 17?)
I have worked this out as the term rule is term + 4.
The next bit has thrown me:
b. What is the position to term rule? (how do you get from 1 to 9, 2 to 13?)
The position to term rule is (position number x ......)
Completely lost, can anybody help and advise on how I can explain this to my son?
Many thanks in advance.
The term-to-term rule (add 4) gives you a finite difference of 4. There's another name for this I think, but can't remember (side effects of hari-kari and all...)
So you have a linear equation with slope 4.
To get from the input to output, multiply by 4 and add ___
Crikey, I am a 44 year old Dad who is completely lost now, many thanks for taking the time and effort to reply but still lost and the linear slope 4, input to output bit confuses my frazzled head more. Perhaps I should have explained I am an idiot at maths and trying to help my 11 year old son! Thanks for your advice though, lost on me but I do appreciate it.
You correctly found the term-to-term rule: to get from one term to the next, add four.
The following will work whenever the term-to-term rule is "add(subtract) a constant". (In our example, the constant is 40).
We can relate the TERM and the POSITION in an equation.
Term = d*Position + c, where d is the term-to-term difference and c is a constant.
So we have
Term = 4*Position + c, since we know the term-to-term difference is 4.
Then we can find c by using ANY of the pairs (position, term).
For example, let's use the pair (1, 9)
9 = 4*1 + c
Solving this for c gives c = 5
So our rule is
T = 4P + 5
Hey thank you, thats clear, I have just done the next 5 questions ready for me to explain and let him do them. Got to the next one though which is different:
Position 1 2 3 4
Term 3 5 7 9
a. What is the term to term rule? (how do you get from 9 to 13, 13 to 17?)
To me the term to term rule is +2 which is what the first part of the question asks but the bit in brackets suggests a +4 which is not what the sequence is doing?
b. What is the position to term rule? (how do you get from 1 to 9, 2 to 13?)
The position to term rule is (position number x....)
Again, much appreciate your help!
Apparently, it is the phrase "position to term rule" that is bothering you. They are asking for the function that assigns to each "position" (1, 2, 3, 4) its corresponding "term" (9, 13, 17, 21). In other words, you want a function, f(n), so that f(1)= 9, f(2)= 13, f(3)= 17, and f(4)= 21.
You know that when n=1, you want f(n)= 9 and that for each unit increase in n, f increases by 4. Okay, look at f(n)= 9+ 4(n-1). When n is 1, n-1= 0 so that is 9 and n-1 counts the number of steps upward and 4(n-1) is just 4 added for each of those steps.
If you had graphed those points ((1, 9), (2, 13), (3, 17), (4, 21)) as Prove It suggested, you would have seen that they lie along a straight line- the function must be "linear", of the form f(x)= ax+b. When x= 1, f(x)= 9 so 9= a+ b. When x= 4, f(x)= 21 so 21= 4a+ b. Subtracting the first equation from that, 3a= 12 so a= 4 (which was what you got before). Putting that into 9= a+ b, we have 9= 4+ b so b= 5. The "rule" (function) is f(n)= 4n+ 5 which is exactly the same as f(n)= 9+ 4(n-1)= 9+ 4n- 4= 4n+ 5.