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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 3 interactions.
If there are 3 children, there can be 12 interactions.
If there are 4 children, there can be 39 interactions.

Which recursive equation represents the pattern?

an = an – 1 + 3(n – 1)

an = an – 1 + 3(n – 1)

an = an – 1 + (3n – 1)

an = an – 1 + (n – 1)3

Respuesta :

aencabo
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:

[tex]a_n=a_{n-1}+3^{n-1}[/tex]

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,

[tex]a_1=0, a_2=3, a_3=12, a_4=39[/tex]

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

[tex]a_2=3\Rightarrow 0+3=a_1+3[/tex]

[tex]a_3=12\Rightarrow 3+9=a_2+3^2[/tex]

[tex]a_4=39\Rightarrow 12+27=a_3+3^3[/tex]

Similarly,

[tex]a_n=a_{n-1}+3^{n-1}[/tex]

Therefore, the required recursive formula is [tex]a_n=a_{n-1}+3^{n-1}[/tex].

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