Arithmetic progression (A.P.) - first term $a$, common difference $d$

$$T_n = a + (n - 1)d \qquad S_n = \frac{n}{2}\left[2a + (n - 1)d\right] = \frac{n}{2}(a + l)$$ where $l$ is the last term.

Geometric progression (G.P.) - first term $a$, common ratio $r$

$$T_n = ar^{\,n-1} \qquad S_n = \frac{a(r^n - 1)}{r - 1} = \frac{a(1 - r^n)}{1 - r}\ (r \neq 1)$$ $$S_\infty = \frac{a}{1 - r}\quad\text{for } |r| < 1$$

Useful

• Common difference $d = T_{n+1} - T_n$; common ratio $r = \dfrac{T_{n+1}}{T_n}$.
• $T_n = S_n - S_{n-1}$.