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Type I - Lacsap's Fractions


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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:

Hello, i got some trouble understanding what you wrote...

when you say

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.
would the table be like this:

n | denominators

2| 2

3| 4

4| 7

5| 11

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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:

Hey, first of all thanks for helping everyone out. I found my general statement in the form of quadratic equation for my numerator and I found the pattern for the denominator, working on the general statement right now, my IA is due at the end of the day so some help would be much appreciated, I looked through all your posts but I didn't understand what you meant by

"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."

Assuming this is for the question En® I have already found the numbers till En(5) but I don't understand what you mean by get a table of the denominators and the n.

Thanks Again.

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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:

Hi... This post of yours really helped me... and i found my statements for a,b and c separately... and they work for all the numbers... however i don't know how to combine it and henceforth express it in terms of r and n... please help... i really need it.... only have few hours left until deadline... please help....

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Haha sorry my explanation in post #16 was not clear enough. I was about to leave to school that's why I didn't have time to explain in long sentences. But I am sick, so I am now at home, not attending school lol. Anyway, look at the picture below:

oYy4W.jpg

Look at all the terms in the blue area and green area. The differences are all 2 (never mind the 11). The differences from the terms in the green area and red area are also both 3 (never mind the 9).

If you look at it from the left side, it's also the same. Look at the picture below:

q91jN.jpg

Same as above right? The terms differences between the 2nd and 3rd terms are all the same and the differences between the 3rd and 4th terms are both same too.

I think this should answer all dany94's, Trikey's and Jyoti's questions (posts #17, #18 and #22). If you are still in doubt just ask again!


What is meant by stating "plot En(1) against n"...what is 'against n' referring to?

Sorry, plot the denominator of En(1) against n is the same as Plot y against x. Meaning denominator of En(1) (i.e. 2, 4, 7, 11) is on the y-axis and n (2, 3, 4, 5) is on the x-axis biggrin.gif


You did? How? I figured out the rest of En2, but what good does that do? How can putting it into a sequence help? and can you hint me on how to start En(3)?

The pattern is exactly the same. I think what I explained above should help, but if you still don't get it please do ask, I don't mind re-explaining in a different way wink.gif

Okay so I know this is a long time after you explained it, but I am actually doing this for my IA for 2012-2013 and I just really wanted to thank you for your explanation on how to find the GS for the denominators. It was very clear and extremely helpful. You're awesome!! :D :D :D

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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:

Hello, i got some trouble understanding what you wrote...

when you say

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.
would the table be like this:

n | denominators

2| 2

3| 4

4| 7

5| 11

I'm confuse on how to put r=2 into a table.

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.

That part is where I am confuse. the (n starts from 4) confuses me a lot.

Edited by Mae Sansebastian
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