So I am looking at ordered pairs: (1,0), (2,3), (3,9), (4,18) , all of the form (x,y)
the second differences of the y values is constant, suggesting a quadratic relationship.
let y = ax^2 + bx + c
at (1,0) ---> a + b + c = 0 ,#1
at (2,3) ---> 4a + 2b + c = 3 , #2
at (3,9) ---> 9a + 3b + c = 9 , #3
subtract #1 from #2 ----> 3a + b = 3
subtract #2 from #3 ----> 5a + b = 6
now subtract those two:
2a = 3
a = 3/2
sub into 3a + b = 3 to find b = -3/2
subbing those back into a+b+c=0 , we get c = 0
so y = (3/2)x^2 - (3/2)x = 3x(x - 1)/2
testing for the data value we did not use,
if x = 4, y = 12(3)/2 = 18 , ok then!!!
when x = 9 , (nine children)
y = 27(8)/2 = 108 <------ number of interactions for 9 children
a playground is being designed where children can interact with their friends in certain combinations. If there is one child, there can be zero interactions. If there are two children, there can only be three interactions. If there are three children, there can only be nine interactions. If there are four children there can be 18 interactions. How many interactions will there be for nine children?
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