Trouble finding general term of a sequence

[Dirtyheads]

Hi,

I'm in college in Québec with the class calculus 2, i am actually learning about sequences and series.

I have a situation where i have to find the general term of a sequence and then say if it converges or diverges, but i am actually not able to even find the general term of that sequence...
Here is the sequence: 5/3 ; 4/3 ; 11/12 ; 7/12 ; 17/48 ; ....

Any help the find the sequence would be really helpful!

Thank you

 

[stapel]

I have a situation where i have to find the general term of a sequence and then say if it converges or diverges, but i am actually not able to even find the general term of that sequence...
Here is the sequence: 5/3 ; 4/3 ; 11/12 ; 7/12 ; 17/48 ; ....

When nothing is obvious with a fractional sequence, one trick can be to convert everything to a common denominator. It may be that the pattern is hiding in what was cancelled out when some of the fractions were reduced. Let's try that here:

. . . . .5/3, 4/3, 11/12, 7/12, 17/48 = 80/48, 64/48, 44/48, 28/48, 17/48

Does this lead anywhere? (Dunno, and I gotta run. I'll check back later.)

 

[Dirtyheads]

When nothing is obvious with a fractional sequence, one trick can be to convert everything to a common denominator. It may be that the pattern is hiding in what was cancelled out when some of the fractions were reduced. Let's try that here:

. . . . .5/3, 4/3, 11/12, 7/12, 17/48 = 80/48, 64/48, 44/48, 28/48, 17/48

Does this lead anywhere? (Dunno, and I gotta run. I'll check back later.)

Thanks i just tried that! but i still can't find anything obvious... I've tried to find the power of this sequence but calculating the difference between each, but there is not pattern i could find... And the last number is really making it harder, since it's the only one that is not an even number...

i really think that it is: *something*/48...

 

[Dirtyheads]

When nothing is obvious with a fractional sequence, one trick can be to convert everything to a common denominator. It may be that the pattern is hiding in what was cancelled out when some of the fractions were reduced. Let's try that here:

. . . . .5/3, 4/3, 11/12, 7/12, 17/48 = 80/48, 64/48, 44/48, 28/48, 17/48

Does this lead anywhere? (Dunno, and I gotta run. I'll check back later.)

Thanks for reminding me of that! I did try it, but it didn't find anything relevant...

I tried to find the power of the general term, by doing the difference between each numbers, but i didn't find a pattern... And the last number of this sequence is really making it harder, since it's an odd number with all the others that are even...

What i am pretty sure now is that it must be *something*/48!

 

[Ishuda]

Hi,

I'm in college in Québec with the class calculus 2, i am actually learning about sequences and series.

I have a situation where i have to find the general term of a sequence and then say if it converges or diverges, but i am actually not able to even find the general term of that sequence...
Here is the sequence: 5/3 ; 4/3 ; 11/12 ; 7/12 ; 17/48 ; ....

Any help the find the sequence would be really helpful!

Thank you

This is probably 'going to be fun'. I don't see anything obvious and, anyway as I'm fond of saying, choose a random number as the next number and you can make up a rational rule to explain the sequence, i.e. 1, 2, 327, ... is described by
a_n = 1 + (n-1) { 1 + [n-2] [ 162 + b (n-3) ] }
for any b you choose.

But back to the problem. Another trick is to do denominator and numerator separately:

Denominator first: Start with two 3's and for succeeding denominators multiple the last minus 1 by 4. That is the 3rd denominator is 4 times the 1st denominator, the 4th denominator is 4 times the 2nd denominator, the 5th denominator is 4 times the 3st denominator, ... Or, letting d_j be the j^th terms denominator,
d₁ = d₂ = 3; d_j = 4 d_j-2 , j = 3, 4, 5, ...

Numerator next: Since the denominator depended on the second denominator back, look at the numerator that way [man is a rationalizing animal, NOT necessarily a rational animal]. O.K. 5, 11, 17 is a difference of six. 4, 7 is a difference of 3. So
n₁ = 5; n_2j+1 = n_2j-1 + 6, j = 1, 2, 3, ...
n₂ = 4; n_2j+2 = n_2j + 3, j = 1, 2, 3, ...

Is that correct? Well I wouldn't bet on it but I think it will do until something better comes along. Oh, and convergence? Yes, to zero since the numerator is basically an arithmetic sequence and the denominator is a geometric sequence with r>1, the denominator will eventually dominate the numerator.

 

[stapel]

The Online Encylclopedia of Integer Sequences has only one matching sequence, and I'm fairly certain that it's not relevant.

In a pinch, one can always find a polynomial passing through the five points, (1, 80), (2, 64), (3, 44), (4, 28), and (5, 17). But there's little guarantee that the output for n = 6 would be a whole number.

There is a French-language result, and it's the only other instance I can find. The pattern proposed there seems to be based on the following restatement of the fractions:

. . . . .5/3, 8/6, 11/12, 14/24, 17/48

Each of the numerator and denominator then has its own pattern.

 

[Dirtyheads]

Wow that was great help from both of you! i really dont think i would have found that all by myself!
You guys are amazing!

Thanks you very much