Question

A playground is being designed where children can interact with their friends in certain combinations.

If there is 1 child, there can be no interactions.
If there are 2 children, there can be only 1 interaction.
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?
an-an-1 +31 - 2
an-an-+ 21 - 3
an = an- 1 + (3n - 2)
an = an- 1 + (2n-3)

Answer 1

The correct recursive equation representing the pattern of interactions among children on a playground is an = an-1 + (3n - 2), revealing that the interactions follow a sequence where each term is related to the previous term plus three times the number of children minus two.

Step-by-step explanation:

The student is trying to find a recursive equation that represents the number of interactions between children in a playground based on the given pattern of interactions. The pattern suggests that each time a child is added, the number of interactions increases by a certain amount that is dependent on the number of children already present. Observing the increase, we can deduce a way to go from one term to the next.

For 1 child, there are 0 interactions. For 2 children, there is 1 interaction. For 3 children, 4 interactions and for 4 children 9 interactions. Looking closely, we see that the number of interactions for n children can be expressed as (n - 1)2. This suggests that the pattern of interactions follows the sequence where each term an is related to the previous term an-1 plus the square of n - 1.

Therefore, the recursive equation representing this pattern is an = an-1 + (3n - 2), where an represents the number of interactions with n children and an-1 represents the number of interactions with n - 1 children.