Find the general term for the sequences: 1, 1, 2, 6, 24, 120, ...

tangosukha said:
kinda stumped on these

1,1,2,6,24,120,...

1/3,-1/8,1/15,-1/24,1/35,...
Click to expand...

These can have multiple solutions:

1/3,-1/8,1/15,-1/24,1/35,..

(-1)^(1-1)/[1*(1+2)], (-1)^(2-1)/[2*(2+2)], (-1)^(3-1)/[3*(3+2)], (-1)^(4-1)/[4*(4+2)], (-1)^(5-1)/[5*(5+2)]....

Now you try the first sequence...
 
tangosukha said:
kinda stumped on these

1,1,2,6,24,120,...

1/3,-1/8,1/15,-1/24,1/35,...
Click to expand...
As Subhotosh Khan implied, finite sequences can have many extensions. However, there are some techniques for 'reducing the sequence' to see if maybe you can make it easier. Two that I use the most are:

Example (1), look at
1, 1, 2, 3, 5, 8, 13, ...
If make a new sequence by subtracting the last term we would start with the second term since there is nothing to subtract for the first term to get
0, 1, 1, 2, 3, 5, ...
Well, if we drop the 0, that's just the first sequence, so you can eventually work out that the Fibonacci sequence can then be described by:
a1 = a2 = 1
an = an-1 + an-2, n>2

Example (2), Look at
1, 1, 4, 36, 576, 14400,
If we make a new sequence by dividing by the last term we get
1, 4, 9, 16, 25
which is just the sequence of squares. So
a0=1
an = n2 an-1

One of those might work for your series.
 
I appreciate the help from all of you, but...

I realized that first seqence listed are factorials after looking ahead in my textbook. I was not taught about factorials and how to write the general term recursively yet. How would I write the general term of that sequence explicitly and not using factorial symbol.
 
tangosukha said:
I realized that first seqence listed are factorials after looking ahead in my textbook. I was not taught about factorials and how to write the general term recursively yet. How would I write the general term of that sequence explicitly and not using factorial symbol.
Click to expand...
By using the definition:

1, 1, 1*2, 1*2*3, 1*2*3*4, ... , 1*2*...*(n-1)

...for a sequence of "n" terms (because the first term is actually zero factorial, so "n" terms goes from 0! to (n-1)!, technically). ;)
 
tangosukha said:
I appreciate the help from all of you, but...

I realized that first seqence listed are factorials after looking ahead in my textbook. I was not taught about factorials and how to write the general term recursively yet. How would I write the general term of that sequence explicitly and not using factorial symbol.
Click to expand...

an = an-2 * (n-1) for n>2
 
tangosukha said:
I appreciate the help from all of you, but...

I realized that first seqence listed are factorials after looking ahead in my textbook. I was not taught about factorials and how to write the general term recursively yet. How would I write the general term of that sequence explicitly and not using factorial symbol.
Click to expand...
Or, another way for the general term
a0 = 1
an = n * an-1; n>0
 
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