Series [Solved!]

X

56, 55, 52, 45 ... ... 

Find the nth term the above series.

Relevant page

<a href="/series-binomial-theorem/series-introduction.php">Series and the Binomial Theorem</a>

What I've done so far

Factors of

56 -   2, 2,      1        14

55 -   1, 1,      5,       11

52 -   2, 2,      1        13

45 -   3, 3,      5

I am really stuck???

X

Is it an arithmetic progression? (How do you know?)

Or is it a geometric one? (How do you know?)

X

I cannot find any order!!!!!!

X

Our process is to decide what kind of progression it is. Can you answer my 4 questions?

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.

X

What is the nth term?

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.

X

Thank you very much

X

There are many  solutions of the nth term for your question such as
       the nth term=  (-1)/3 n^3+n^2-5/3 n+57
or 
       the nth term= 57+n-2^n

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.