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Type I - Lacsap's Fractions


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can someone please guide me to generalise the formulas? i have found different formulas for R=1 until R=5. PLEASE HELP.

they're all quadratic, right? so then just formulate each of a,b,c (in denominator=an^2+bn+c) in terms of r. then combine them.

btw dari sekolah mana? satu sekolah sama Glacierfreeze?

Edited by Desy Glau
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hey,

i really have troubles finding the gs just for the denominator. I read everything but i just cant figure it out, even with the help of some of yours. Is there maybe anotherway how i can find the denominator? and if no, could somebody please try to explain it to me again?

thank you guys

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hey,

i really have troubles finding the gs just for the denominator. I read everything but i just cant figure it out, even with the help of some of yours. Is there maybe anotherway how i can find the denominator? and if no, could somebody please try to explain it to me again?

thank you guys

no to both questions :/

first have you noticed the pattern or not?

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ive tried to do that... go the equation for the first one just with n but no r....is taht wrong? and i cant find the ones for r(2), r(3) and so on

you also written Hint: The general statement is in quadratic form (denominator=an²+bn+c)

does the n represent the row number in your formula?

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ive tried to do that... go the equation for the first one just with n but no r....is taht wrong? and i cant find the ones for r(2), r(3) and so on

you also written Hint: The general statement is in quadratic form (denominator=an²+bn+c)

does the n represent the row number in your formula?

it's right, the equation for each diagonal is in terms of n only. after you get many equations (many diagonals) you generalise further so that the equation is in terms of n and r.

yes... n is always the row number, as written in the task paper, right?

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mmm are you sure it's correct? I've just tried to work out the general statement of En(r ) and I got a quite complicated formula in terms of n and r. it did require some effort to get to the general statement.

there are many ways to get to the GS but I myself used the graphical method (b/c I'm lazy...) but if you know how to find it using another method, go for it. I personally don't favour the graphical method (plotting a graph and finding the best fit curve) for a Math IA, though it's a use of technology :S

The numerator part is very obvious and easy, right? So I don't need to discuss it.

After you found the 6th and 7th rows, draw a new triangle if possible so you can see the patterns more easily.

Remember that 1 is the 1st term in each row, and in this term r=0.

When r=1 (the 2nd term from the left), get a table of the denominators and the n (n starts from 2). Then you find the GS for the denominator in terms of n.

When r=2 (the 3rd term from the left), get a table of the denominators and the n (n starts from 3). Then you find the GS for the denominator in terms of n.

When r=3 (the 4th term from the left), get a table of the denominators and the n (n starts from 4). Then you find the GS for the denominator in terms of n.

Do the same thing for when r=4 and when r=5.

You've got all the general statements in terms of n. Now your goal is to make them to be ONE general statement in terms of r and n.

Hint: The general statement is in quadratic form (denominator=an²+bn+c)

Tabulate r and the general statements. I am sure you will see some kind of pattern in the values of a, b and c. Find them with a similar method as before,

Get a table of a and r. Then you find the GS for a in terms of r.

Get a table of b and r. Then you find the GS for b in terms of r.

Get a table of c and r. Then you find the GS for c in terms of r.

So you've got a, b and c. Put them in the general statement involving n. You will get the GS for the denominator in terms of n and r.

This task is fairly easy, though it seems complicated in the beginning. Good luck to all of you!! :gluck:

This was the most helpful post for me. Thank you very much, even though I had figured out these patterns, you really helped me "materialize" it. This is probably the best method of finding the general statement, as it provides plenty of opportunities to use technology. Again, thanks :)
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