Answer:
F(n) = n^2+4n+2
This function produces the number of small squares, F(n), per figure number n.
Step-by-step 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
