Question

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

Answer 1

From your situations given:
If there is 1 child, there can be 0 interactions. 
If there are 2 children, there can be 3 interactions. 
If there are 3 children, there can be 12 interactions. 
If there are 4 children, there can be 39 interactions. 

The equation would be an = an – 1 + 3(n – 1)

Answer 2

Answer:

$a_n=a_{n-1}+3^{n-1}$

Step-by-step explanation:

It is given that

If there is 1 child, there can be 0 interactions.

If there are 2 children, there can be 3 interactions.

If there are 3 children, there can be 12 interactions.

If there are 4 children, there can be 39 interactions.

It means we have,

$a_1=0, a_2=3, a_3=12, a_4=39$

We need to find the recursive equation that represents the pattern.

$a_2=3\Rightarrow 0+3=a_1+3$

$a_3=12\Rightarrow 3+9=a_2+3^2$

$a_4=39\Rightarrow 12+27=a_3+3^3$

Similarly,

$a_n=a_{n-1}+3^{n-1}$

Therefore, the required recursive formula is $a_n=a_{n-1}+3^{n-1}$.