The pattern is adding 2 stars on top of the previous figure, so figure number 4 would look like this:
* *
* *
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* *
Since the pattern is adding 2 stars every time, the explicit formula (for the number of stars) would be
[tex]f(n)=2n[/tex]
Since when n=1, i.e. figure 1, we have 2*1=2 stars
when n=2, i.e. figure 2, we have 2*2=4 stars
when n=3, i.e. figure 3, we have 2*3=6 stars
and so on.
The recursive formula lays on the fact that we add 2 stars to the previous figure, which means that the fomula is
[tex]f_n = f_{n-1}+2[/tex]
In fact, what we just wrote is simply "the number of stars at a given iteration is the number of stars at the previous iteration, plus 2.
