The General (nth) Term of an AP and a GP

Once a progression's first term and common difference/ratio are known, its general term (the $n$-th term, $T_n$) can be written directly as a formula, without listing every term up to it.

Both formulas say the same kind of thing: to reach the $n$-th term from the first term, apply the "step" operation exactly $(n-1)$ times — add $d$ a total of $(n-1)$ times for an AP, or multiply by $r$ a total of $(n-1)$ times for a GP (not $n$ times, since the first term $T_1=a$ itself needs zero steps).

Example (AP): for $3,7,11,15,\ldots$ ($a=3,d=4$), the $10$th term is $T_{10}=3+(10-1)(4)=3+36=39$.

Example (GP): for $2,6,18,54,\ldots$ ($a=2,r=3$), the $6$th term is $T_6=2\times3^{5}=2\times243=486$. …