how is the equation of the geometric series be an=a1*r^(n-1)

$$a_n=a_1*r^{n-1}$$

 

You mean why is it?

Let's look at a GP

 

$$The first term is $a_1$\\
The second term is $a_1*r$\\
The third term is the second term *r = $a_1*r*r = ar^2$\\
The forth term is the third term *r = $a_1r^2*r = ar^3$\\
following the pattern,
The twenieth term is going to be $ a_1r^{19}$\\
and\\
The nth term is going to be $ a_1r^{n-1}$\\\\
I hope that that helps.$$

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