Now I cannot find the nth terms. The reason is that I have gone wrong in my analysis.

X

The series has a decreasing tendency.
The difference between the figures are:
-1, -3, -7
(-5 is missing in the range)
Say it is -1, -3, -7, -11, -17, -19
there is a jump over every 3rd number.
The series is 56, 55, 52, 45, 34, 17, -2
As such it is an AP.
As for geometric progression(GP), the figures do no have a common ratio as shown below and hence is not a GP.
56 55 52 45
55/56 52/55 45/52
0.982142857 0.945454545 0.865384615
It looks to be more an AP than a GP.
Now I cannot find the nth terms. The reason is that I have gone wrong in my analysis.

You're right that it seems to be more like an AP, but it's not, and neither is your example, 56, 55, 52, 45, 34, 17, -2, since there is no common difference.

Please also note it's not a series (where the terms are added) - it's a progression (where there is some common pattern between each of the terms).

I believe in this case there is not enough information. When it's neither AP nor GP, we usually need more terms to decide what's going on. Sometimes people make up some pattern that seems consistent to them, but I don't believe that is the case here.

X

You're right that it seems to be more like an AP, but it's not, and neither is your example, 56, 55, 52, 45, 34, 17, -2, since there is no common difference.
Please also note it's not a series (where the terms are added) - it's a progression (where there is some common pattern between each of the terms).
I believe in this case there is not enough information. When it's neither AP nor GP, we usually need more terms to decide what's going on. Sometimes people make up some pattern that seems consistent to them, but I don't believe that is the case here.

So both expressions look good! Thanks for your inputs, Majid67.

X

@Majid67: It appears we gave up too easily.
Original sequence: 56, 55, 52, 45, ...
Using your first proposed expression,
nth term `= -1/3 n^3+n^2-5/3 n+57`
When `n=1,`
`-1/3 (1)^3+(1)^2-5/3(1)+57 ` `= (-1)/3+1-5/3+57 = 56` (OK)
When `n=2,`
`-1/3 (2)^3+(2)^2-5/3(2)+57 ` `= (-8)/3+4-10/3+57 = 55` (OK)
When `n=3,`
`-1/3 (3)^3+(3)^2-5(3/3)+57 ` `= (-27)/3+9-15/3+57 = 52` (OK)
When `n=4,`
`-1/3 (4)^3+(4)^2-5(4/3)+57 ` `= (-64)/3+16-20/3+57 = 45` (OK)
Using your second proposed expression,
nth term `= 57+n-2^n`
When `n=1,`
`57+1-2^1 = 57+1-2 = 56` (OK)
When `n=2,`
`57+2-2^2 = 57+2-4 = 55` (OK)
When `n=3,`
`57+3-2^3 = 57+3-8 = 52` (OK)
When `n=4,`
`57+4-2^4 = 57+4-16 = 45` (OK)
So both expressions look good! Thanks for your inputs, Majid67.