Remark 2.2.3.

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.