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 1 interactions.
If there are 3 children, there can be 4 interactions.
If there are 4 children, there can be 9 interactions.
Which recursive equation represents the pattern?
Thank you
Could it be written as a(n)=a(n-1)+2^n-3
To find the recursive equation that represents the pattern of interactions, we need to analyze the given information.
First, let's list the number of interactions for each number of children:
- 1 child: 0 interactions
- 2 children: 1 interaction
- 3 children: 4 interactions
- 4 children: 9 interactions
We can observe that the number of interactions for each case is increasing by adding a squared number. For example, when there are 2 children, there is 1 interaction (1^2), and when there are 3 children, there are 4 interactions (2^2). Similarly, when there are 4 children, there are 9 interactions (3^2).
Based on this observation, we can conclude that the recursive equation representing the pattern is:
𝐈(𝑛) = 𝐈(𝑛 − 1) + 𝑛^2
Where 𝐈(𝑛) represents the number of interactions for 𝑛 children, and 𝑛 represents the number of children. This equation states that the number of interactions for 𝑛 children is equal to the number of interactions for 𝑛-1 children plus 𝑛^2 (the square of the number of children).
a(1) = 0
a(n+1) = a(n)+(2n-1)
or,
a(n) = (n-1)^2
Did you even check out your idea?
a(1) = 0
a(2) = a(1) + 2^2-3 = 0+1 = 1
a(3) = a(2) + 2^3-3 = 1+5 = 6
a(4) = a(3) + 2^4-3 = 6+13 = 19