Arith $n$th

$a_n=a_1+(n-1)d$

First term + $(n-1)$ jumps of $d$

Arith sum

$S_n=\frac{n(a_1+a_n)}{2}$

Average × count

Geom $n$th

$a_n=a_1\cdot r^{n-1}$

First term × ratio $(n-1)$ times