Geometric Progression

At a glance 11 formulas

1. 1nth term$T_n = a\,r^{\,n-1}$$a = \text{first term},\ r = \text{common ratio}$
2. 2Sum of n terms$S_n = a\,\dfrac{r^n - 1}{r - 1},\quad r \ne 1$$\text{doubling from 1: } 1 + 2 + \dots + 2^{n-1} = 2^n - 1$
3. 3Three in a row$b^2 = ac$$a, b, c \text{ consecutive; in an AP: } b = \tfrac{a + c}{2}$
4. 4Three unknown terms$\dfrac{a}{r},\ a,\ ar$$\text{product} = a^3 \text{, so a given product fixes } a$
5. 5The infinite GP$S_\infty = \dfrac{a}{1 - r},\quad |r| < 1$$|r| \ge 1 \text{: the sum never settles}$
6. 6Alternating signs$a - ar + ar^2 - \dots = \dfrac{a}{1 + r}$$0 < r < 1$
7. 7Negative ratioWith $r < 0$ the running total swings above and below $S_\infty$, so a partial sum can be bigger than the whole infinite sum.
8. 8Bouncing ball$\text{distance} = h\,\dfrac{1 + r}{1 - r}$$h = \text{drop height},\ r = \text{rebound fraction}$
9. 9Blocks of k terms$S_{2k} = S_k\,(1 + r^k)$$\text{the next } k \text{ terms} = r^k \times \text{the first } k$
10. 10Repeating patternIf the signs or terms repeat every $k$ terms, add each group of $k$: the groups form a GP with ratio $r^k$.
11. 11Pairs from the ends$T_k \times T_{n+1-k} = a\,l$$S \div R = al\ (S = \text{sum},\ R = \text{sum of reciprocals}),\ \text{product} = (al)^{n/2},\ l = \text{last term}$