# Homework Help: What are the sub sequances here

1. Dec 12, 2008

### transgalactic

1/2 , 1/3 , 2/3 , 1/4 , 2/4 , 3/4 , 1/5 ,2/5 ,3/5 ,4/5 ...

i cant see the patterns here?

2. Dec 12, 2008

### mutton

There are infinitely many subsequences of any sequence: just delete some elements.

For the pattern, first look at the denominators only, and then look at how the numerators change for each denominator.

3. Dec 12, 2008

### transgalactic

ok i see numbers
1/2 2/3 3/4
a1 a3 a6
but they dont have constant gap between then

so i cant do a formula for that

Last edited: Dec 12, 2008
4. Dec 12, 2008

### nweibley

I see two things going on here... the denominator increases, and with each new denominator d a sum of 1/d to (d-1)/d fractions. If you are familiar with programming this sequence would easily be described with a nested loop.

That should get you started ;)

5. Dec 12, 2008

### transgalactic

i see that too but i need to break into sUb sequences

what you say doesnt help me with that

6. Dec 12, 2008

### nweibley

As previously stated there are infinite subsequences in this sequence... is the original question more specific?

7. Dec 12, 2008

### transgalactic

ok i need at least two
and their location n(k) formula

8. Dec 12, 2008

### mutton

To describe the sequence 1, 3, 6, 10, 15, ...,

n(1) = 1
n(2) = n(1) + 2
n(3) = n(2) + 3
...
n(k) = ?

If you are familiar with Pascal's triangle, you can get a "nicer" expression from it.

9. Dec 12, 2008

### transgalactic

this is a recursion
i need linear n(k) formula

10. Dec 12, 2008

### mathlete

edit:

Take the numbers with numerator 1. They clearly form the sub-sequence 1/(k+1) k=1...∞

Now, these occur at:
1/2 -> position 1
1/3 -> position 2
1/4 -> position 4
1/5 -> position 7
1/6 -> position 11
1/n -> position ??

Hint: the distance between positions increases by one every time
Double hint: ½n² + ½n + 1

Last edited: Dec 12, 2008
11. Dec 12, 2008

### HallsofIvy

One thing I see is 1/2, 2/3, 3/4, 4/5, ... which clearly converges
Another is 1/3, 1/4, 1/5, ... which also converges.

As has been pointed out there are an infinite number of subsequences in any sequence. Perhaps a better question would be, "What are the subsequential limits?"

12. Dec 13, 2008

### transgalactic

i need to find the sub sequences in order to find their limits
so there may be endless number of limits

unless this series converges
in which case all of the sub sequences limits equal
the limit of the sequence

what to do in this case:
1/2 , 1/3 , 2/3 , 1/4 , 2/4 , 3/4 , 1/5 ,2/5 ,3/5 ,4/5 ...

??