# What are the sub sequances here

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?

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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.

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

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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 ;)

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

what you say doesnt help me with that

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

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

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
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.

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

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

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HallsofIvy
Homework Helper
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?
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?"

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 ...

??