11.3 Geometric Sequences

Geometric Sequences

Geometric sequences are built on a simple idea: you get each term by multiplying the previous one by the same constant factor. That constant factor, called the common ratio, controls everything about how the sequence behaves. These sequences show up whenever you're modeling exponential growth or decay, from compound interest to radioactive half-lives.

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Common Ratio in Geometric Sequences

The common ratio ($r$) is the constant factor you multiply by to move from one term to the next. You can find it by dividing any term by the one before it:

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

This works for any consecutive pair of terms in the sequence. If the ratio isn't the same between every pair, you're not dealing with a geometric sequence.

The value of $r$ tells you exactly how the sequence will behave:

• $|r| > 1$: The terms grow exponentially. For example, $r = 3$ gives you 2, 6, 18, 54, ...
• $0 < |r| < 1$: The terms shrink toward zero (exponential decay). For example, $r = \frac{1}{2}$ gives you 8, 4, 2, 1, 0.5, ...
• $r = 1$: Every term is the same. The sequence is constant.
• $r = -1$: The terms alternate between two values, like 5, -5, 5, -5, ...
• $r < -1$ or $-1 < r < 0$: The terms alternate in sign and grow or decay in magnitude. For instance, $r = -2$ gives 1, -2, 4, -8, 16, ... (alternating and growing), while $r = -\frac{1}{2}$ gives 8, -4, 2, -1, ... (alternating and decaying).

Generating Geometric Sequence Terms

Start with the first term $a_1$ and multiply by $r$ repeatedly:

• Second term: $a_2 = a_1 \cdot r$
• Third term: $a_3 = a_1 \cdot r^2$
• Fourth term: $a_4 = a_1 \cdot r^3$

Notice the pattern: the exponent on $r$ is always one less than the term number. This gives us the general formula for the $n$th term:

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

That exponent being $n - 1$ (not $n$) is a common source of errors on tests. The first term has $r^0 = 1$, so $a_1$ stays unchanged, which makes sense.

Recursive Formulas for Geometric Sequences

A recursive formula defines each term using the previous term:

$a_{n+1} = a_n \cdot r, \quad n \geq 1$

To use this, you need two pieces of information: the first term $a_1$ and the common ratio $r$. Then you build the sequence one term at a time.

Example: Given $a_1 = 3$ and $r = 2$, find $a_2, a_3, a_4$.

1. $a_2 = a_1 \cdot r = 3 \cdot 2 = 6$
2. $a_3 = a_2 \cdot r = 6 \cdot 2 = 12$
3. $a_4 = a_3 \cdot r = 12 \cdot 2 = 24$

Recursive formulas are great when you're extending a sequence term by term, but they're slow if you need, say, the 50th term. That's where explicit formulas come in.

Explicit Formulas for Sequence Terms

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

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

You need three things: the first term $a_1$, the common ratio $r$, and the position $n$ of the term you want.

Example: Find the 10th term of a geometric sequence with $a_1 = 2$ and $r = 3$.

$a_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9 = 2 \cdot 19{,}683 = 39{,}366$

Try doing that recursively and you'll see why the explicit formula is so useful for large $n$.

Sequence Behavior and Series

Convergence vs. Divergence describes what happens to the terms as $n$ gets very large:

• When $|r| < 1$, the terms shrink toward zero. The sequence converges.
• When $|r| \geq 1$, the terms either grow without bound or keep oscillating with constant/increasing magnitude. The sequence diverges.

Partial sums add up a finite number of terms. The formula for the sum of the first $n$ terms is:

$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1$

Infinite geometric series add up all the terms in the sequence. This only produces a finite answer when the sequence converges ($|r| < 1$). In that case:

$S_{\infty} = \frac{a_1}{1 - r}$

If $|r| \geq 1$, the infinite series diverges and has no finite sum.

Geometric vs. Arithmetic: Don't confuse these two. Arithmetic sequences add a constant difference between terms; geometric sequences multiply by a constant ratio. The formulas look different, and they model very different types of growth.