13.3 Geometric Sequences

Geometric Sequences

Geometric sequences are patterns where each term is found by multiplying the previous term by a fixed number called the common ratio. They show up anywhere you see repeated multiplication: compound interest, population growth, radioactive decay, and more.

These sequences connect directly to exponential functions, so getting comfortable with them now pays off when you hit exponential and logarithmic models later in the course.

Geometric Sequences

Common ratio in geometric sequences

A geometric sequence is a sequence where each term equals the previous term multiplied by a fixed, non-zero number called the common ratio ($r$). To find the common ratio, divide any term by the term right before it:

$r = \frac{a_{n+1}}{a_n}$

The common ratio can be positive, negative, or a fraction:

• Positive $r$: 2, 4, 8, 16 ($r = 2$)
• Negative $r$: $-3, 6, -12, 24$ ($r = -2$). Notice the signs alternate.
• Fractional $r$: 1, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$ ($r = \frac{1}{2}$)

The size of $|r|$ tells you whether the terms grow or shrink:

• If $|r| > 1$, terms increase in absolute value: 2, 6, 18, 54 ($r = 3$)
• If $|r| < 1$, terms decrease in absolute value: 80, 40, 20, 10 ($r = \frac{1}{2}$)
• If $|r| = 1$, every term is the same (or alternates sign if $r = -1$)

Term generation for geometric sequences

To build a geometric sequence, start with the first term $a_1$ and keep multiplying by $r$:

• $a_2 = a_1 \cdot r$
• $a_3 = a_1 \cdot r^2$
• $a_4 = a_1 \cdot r^3$

See the pattern? The exponent on $r$ is always one less than the term number. That gives you the general formula for the $n$th term:

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

A couple of examples to see this in action:

• Sequence: 3, 6, 12, 24, ... Here $a_1 = 3$ and $r = \frac{6}{3} = 2$, so $a_n = 3 \cdot 2^{n-1}$.
• Sequence: 1000, 100, 10, 1, ... Here $a_1 = 1000$ and $r = \frac{100}{1000} = \frac{1}{10}$, so $a_n = 1000 \cdot \left(\frac{1}{10}\right)^{n-1}$.

Recursive formulas for sequence analysis

A recursive formula defines each term using the term right before it. For a geometric sequence, it looks like this:

$a_n = a_{n-1} \cdot r, \quad \text{with } a_1 \text{ given}$

You always need two pieces of information: the first term and the common ratio. From there, you generate terms one at a time.

Example: $a_1 = 5$, $r = 3$

1. $a_2 = 5 \cdot 3 = 15$
2. $a_3 = 15 \cdot 3 = 45$
3. $a_4 = 45 \cdot 3 = 135$

Recursive formulas are great when you need the next few terms or want to spot a pattern. The downside is that finding, say, the 50th term requires computing all 49 terms before it. That's where the explicit formula comes in.

Explicit formulas for specific terms

The explicit formula lets you jump straight to any term without calculating all the ones before it:

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

• $a_1$ = first term
• $r$ = common ratio
• $n$ = position of the term you want

Example: Find $a_5$ when $a_1 = 2$ and $r = 4$.

1. Substitute into the formula: $a_5 = 2 \cdot 4^{5-1}$
2. Simplify the exponent: $a_5 = 2 \cdot 4^4$
3. Calculate: $a_5 = 2 \cdot 256 = 512$

This is especially useful for terms far into the sequence. Finding $a_{100}$ with a recursive formula would take 99 steps, but the explicit formula gets you there in one calculation.

Connections to exponential functions and applications

The explicit formula $a_n = a_1 \cdot r^{n-1}$ looks a lot like an exponential function $f(x) = a \cdot b^x$, and that's not a coincidence. A geometric sequence is really just an exponential function evaluated at whole-number inputs. The common ratio $r$ plays the same role as the base of the exponential.

This connection matters for real-world problems:

• Compound interest: An account earning 5% annually has a common ratio of $r = 1.05$. After $n$ years, the balance follows a geometric pattern.
• Depreciation: A car losing 15% of its value each year has $r = 0.85$, so its value forms a decreasing geometric sequence.
• Biology: A bacteria colony doubling every hour has $r = 2$.

If you need to solve for $n$ (e.g., "how many years until the investment doubles?"), you'll rearrange the explicit formula and use logarithms to isolate the exponent. That's the bridge between this topic and the logarithmic equations you've already studied.