Geometric Progression (GP)

A bacterium splits in two every hour. After $n$ hours, the population has multiplied by $2^n$. The population history $1, 2, 4, 8, 16, \dots$ is a geometric progression: each term is a fixed multiple of the previous. GPs describe every kind of exponential growth (and decay) , interest, radioactive substances, viral spread.

Definitions

A sequence $a_1, a_2, a_3, \dots$ is a geometric progression (GP) if the ratio $a_{n+1} / a_n$ is the same nonzero constant for every $n$. That constant is the common ratio, denoted $r$.

If the first term is $a$, the GP is $a,\ ar,\ ar^2,\ ar^3,\ \dots$

We assume $a \ne 0$. The ratio $r$ can be any nonzero real number , positive, negative, fractional. If $|r| > 1$ the terms grow; if $|r| < 1$ they shrink; if $r = -1$ they alternate.

The $n$-th term

$\boxed{a_n = a r^{n - 1}.}$

This is an exponential function of $n$. GPs are exactly the sequences whose general term is exponential in $n$.

Sum of the first $n$ terms

Let $S_n = a + ar + ar^2 + \dots + ar^{n-1}$. Multiply by $r$: $r S_n = ar + ar^2 + \dots + ar^n$. Subtract: $S_n - r S_n = a - ar^n$, so $(1 - r) S_n = a (1 - r^n)$, giving $\boxed{S_n = a \cdot \frac{1 - r^n}{1 - r} = a \cdot \frac{r^n - 1}{r - 1}, \quad r \ne 1.}$

If $r = 1$, all terms equal $a$, so $S_n = n a$.

Infinite GP

If $|r| < 1$, $r^n \to 0$ as $n \to \infty$. So $S_\infty = \lim_{n \to \infty} S_n$ exists: $\boxed{S_\infty = \frac{a}{1 - r}, \quad |r| < 1.}$

If $|r| \ge 1$ (and $a \ne 0$), the infinite GP has no finite sum.

Properties

Three terms in GP. Often use $a/r, a, ar$. Four terms: $a/r^3, a/r, ar, ar^3$ (common ratio $r^2$).

Geometric mean property. Every middle term squared equals the product of its neighbours: $a_n^2 = a_{n-1} a_{n+1}$.

Multiplication of constants. Multiplying every term by a fixed nonzero constant gives another GP with the same ratio.

Worked examples

Example 1. Find the $7$-th term of $3, 6, 12, 24, \dots$

$a = 3, r = 2$. $a_7 = 3 \cdot 2^6 = 192$.

Example 2. Find the sum of $2 + 6 + 18 + 54 + \dots$ to $10$ terms.

$a = 2, r = 3, n = 10$. $S_{10} = 2 \cdot \dfrac{3^{10} - 1}{3 - 1} = 3^{10} - 1 = 59048$.

Example 3. Find $\sum_{k=0}^{\infty} \dfrac{1}{2^k}$.

This is a GP with $a = 1, r = 1/2$. $S_\infty = \dfrac{1}{1 - 1/2} = 2$.

Example 4. Three numbers are in GP. Their sum is $21$ and product is $216$. Find them.

Let them be $a/r, a, ar$. Product: $a^3 = 216 \Rightarrow a = 6$. Sum: $6/r + 6 + 6r = 21 \Rightarrow 6/r + 6r = 15 \Rightarrow 6 + 6 r^2 = 15 r \Rightarrow 6 r^2 - 15 r + 6 = 0 \Rightarrow 2r^2 - 5r + 2 = 0 \Rightarrow r = 2$ or $r = 1/2$. Numbers: $3, 6, 12$.

Example 5. Convert the repeating decimal $0.\overline{36}$ to a fraction.

$0.\overline{36} = 0.363636\dots = 0.36 + 0.0036 + 0.000036 + \dots$ This is a GP with $a = 0.36 = 36/100$ and $r = 1/100$. Sum $= \dfrac{36/100}{1 - 1/100} = \dfrac{36/100}{99/100} = \dfrac{36}{99} = \dfrac{4}{11}$.

Try it yourself

1. Find the $10$-th term of $5, 10, 20, 40, \dots$
2. Find the sum of $3 + 6 + 12 + \dots$ to $8$ terms.
3. Find the sum of the infinite GP $4, 2, 1, 1/2, 1/4, \dots$
4. Three numbers in GP have product $1000$ and sum $35$. Find them.
5. The $3$rd term of a GP is $24$, the $6$th term is $192$. Find the $9$th term.
6. Express $0.\overline{142857}$ as a fraction. (Hint: it equals $1/7$.)
7. The sum of an infinite GP is $15$ and the sum of squares of its terms is $45$. Find the GP.
8. How many terms of the GP $2, 4, 8, \dots$ sum to $2046$?
9. If the sum of $n$ terms of a GP is $S$ and the product is $P$, prove $P^2 = (a \cdot \text{last})^n$.
10. Find three numbers in GP whose sum is $13$ and sum of squares is $91$.
11. A ball is dropped from height $20$ m. After each bounce it rises to $4/5$ of its previous height. Total distance travelled?
12. Find the sum to $n$ terms: $7 + 77 + 777 + \dots$ (Hint: write as $\dfrac{7}{9}(10 - 1 + 100 - 1 + \dots)$.)

Pitfalls / Tricks

• Use $S_n = a \dfrac{r^n - 1}{r - 1}$ when $r > 1$ and $S_n = a \dfrac{1 - r^n}{1 - r}$ when $|r| < 1$ , keeps the sign clean.
• Infinite GP sum requires $|r| < 1$.
• For three terms in GP, choose $a/r, a, ar$ to make sums and products simple.
• Insight. A GP turns multiplication into addition (in the exponent). Whenever a problem has products or quotients of consecutive terms, GP is in the wings.