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Can someone write the quadratic function from this pattern? Thanks

Can someone write the quadratic function from this pattern? Thanks-example-1

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Answer:

F(n) = n^2+4n+2

This function produces the number of small squares, F(n), per figure number n.

Explanation:

figure 1 :

I see a 2 by 3 rectangle then 1 more square.

figure 2:

I see a 3 by 4 rectangle then 2 more squares.

figure 3:

I see a 4 by 5 rectangle then 3 more squares.

The pattern is that the dimensions I'm seeing for the rectangle is 1 more than the figure number by 2 more than the figure number while also noticing the figure number matches the number of squares not included in said rectangle.

Recall the area of rectangle is base×height.

So for figure n we have the number of squares in all is:

(n+1)(n+2)+n

We could write in standard form by doing some multiplication and adding.

Distribute:

n(n+2)+1(n+2)+n

n^2+2n+1n+2+n

Combine like terms:

n^2+3n+2+n

n^2+4n+2

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