SOLUTION: Hello. I would appreciate an explanation to the following question: A playground is being designed where children can interact with their friends in certain combinations. If

Algebra ->  Sequences-and-series -> SOLUTION: Hello. I would appreciate an explanation to the following question: A playground is being designed where children can interact with their friends in certain combinations. If       Log On


   

Question 495959: Hello. I would appreciate an explanation to the following question:
A playground is being designed where children can interact with their friends in certain combinations.
If there is 1 child, there can be 0 interactions.
If there are 2 children, there can be only 1 interaction.
If there are 3 children , there can be 5 interactions.
If there are 4 children, there can be 14 interactions.
Which recursive pattern represents the pattern?
asubn=asubn-1+2^(n-1)
asubn=asubn+(n-1)^2
asubn=asubn+2(n-1)
asubn=asubn+(2n-1)

So far I have figured out by setting up this basic pattern chart that:

n a
1 0 +1
2 1 +4
3 5 +9
4 14
Is the equation always the same? or does it change with the given information. I got this wrong and would appreciate understanding why? I understand how to make derive the sequence information. What if I need the 50th term? I can easily get the ninth term as in the previous question on this assignment. But having trouble understanding how to write the formula.
I appreciate you help. This is a wonderful resource for students who take math seriously because the tutors answer the questions so thoroughly. Again, thank you in advance.
Justina
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the formula that works with the data is as follows:

a[n] = a[n-1] + (n-1)^2

here's a table of values based on that formula.
the table starts with a[1] being given as 0. 

        n        (n-1)^2         a[n]

        1        n/a            given as 0
        2        1^2 = 1        1 + 0 = 1
        3        2^2 = 4        4 + 1 = 5
        4        3^2 = 9        9 + 5 = 14
        5        4^2 = 16       16 + 14 = 30
        6        5^2 = 25       25 + 30 = 55
        ...


you are given that the a[1] = 0
the rest of them are derived from the formula.
the same formula is used every time.
let me know if this solves the problem for you.