If you look at other textbooks or online, you might find that their closed formulas for arithmetic and geometric sequences differ from ours. Specifically, you might find the formulas \(a_n = a +(n-1)d\) (arithmetic) and \(a_n = a\cdot r^{n-1}\) (geometric). Which is correct? Both! In our case, we take \(a\) to be \(a_0\text{.}\) If instead we had \(a_1\) as our initial term, we would get the (slightly more complicated) formulas you find elsewhere.
in-context