In the equation = (r)^n-1what does the n-1 mean?

I'm pretty sure you mean    where  a₁ is the first term of the geometric sequence and r is the common ratio. the n is the nth term.the n-1 makes sure that the number n is exactly what it means.it basically aligns the exponent to what you want.consider this example. we have the first term a₁ be 3 and the common ratio r be 2. the formula for this geometric sequence is therefore    the first term is 3. since the common ratio is 2, the second term should be 6. the third term should be 12.  the second term, we can set n = 2 and get it. the n - 1 part will subtract 1 from n to get the correct exponent for the second term.    and the third term would be aligned as well for n = 3:     ======now, if the formula were simply  , you would not get the second term if you have n = 2:     12 is supposed to be the third term. so we needed that n-1 to align ourselves to what we want, otherwise we won't be able to say intuitively that "second: n = 2", "third: n = 3",  "fourth: n = 4", etc.