12.3 Geometric Sequences and Series

Geometric sequences are number patterns where each term is found by multiplying the previous term by a fixed number called the common ratio. They show up whenever something grows or shrinks by a constant percentage, which makes them the foundation for modeling compound interest, population growth, and depreciation.

This section covers how to identify geometric sequences, calculate any term using the general formula, find sums of both finite and infinite geometric series, and apply these tools to real-world problems.

Geometric Sequences

Identifying 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 ($r$). For example: 2, 6, 18, 54, ... has a common ratio of 3.

To confirm a sequence is geometric, divide any term by the one before it. If you get the same value every time, that's your common ratio:

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

The behavior of the sequence depends entirely on $r$:

• If $|r| > 1$, terms grow in magnitude: 2, 8, 32, 128, ...
• If $0 < |r| < 1$, terms shrink toward zero: 1, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, ...
• If $r < 0$, terms alternate between positive and negative: 1, -2, 4, -8, ...

A geometric sequence can also be defined recursively, where each term is given in terms of the previous one: $a_{n+1} = a_n \cdot r$.

General Term Calculation

The formula for the nth term of a geometric sequence is:

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

• $a_n$ = the nth term you're looking for
• $a_1$ = the first term
• $r$ = the common ratio
• $n$ = the term number

Notice the exponent is $n - 1$, not $n$. That's because the first term ($n = 1$) has the ratio applied zero times.

Example: Find the 7th term of the sequence 4, 12, 36, 108, ...

1. Identify $a_1 = 4$
2. Find the common ratio: $r = \frac{12}{4} = 3$
3. Plug into the formula with $n = 7$: $a_7 = 4 \cdot 3^{7-1} = 4 \cdot 3^6 = 4 \cdot 729 = 2{,}916$

Geometric mean: If you know two terms in a geometric sequence and need the term between them, the geometric mean gives you that middle value. For two positive numbers $a$ and $b$, the geometric mean is $\sqrt{a \cdot b}$.

Finite Sequence Sums

To add up the first $n$ terms of a geometric sequence, use the partial sum formula:

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

• $S_n$ = sum of the first $n$ terms
• $a_1$ = first term
• $r$ = common ratio
• $n$ = number of terms

Example: Find the sum of the first 5 terms of 3, 9, 27, 81, ...

1. Identify $a_1 = 3$, $r = 3$, $n = 5$
2. Substitute into the formula:

$S_5 = \frac{3(1 - 3^5)}{1 - 3} = \frac{3(1 - 243)}{-2} = \frac{3(-242)}{-2} = \frac{-726}{-2} = 363$

A common mistake here is forgetting that both the numerator and denominator are negative when $r > 1$, so the negatives cancel and the sum is positive.

Infinite Series Evaluation

An infinite geometric series adds up all the terms forever: $a_1 + a_1 r + a_1 r^2 + a_1 r^3 + \cdots$

This only produces a finite sum when the terms shrink toward zero, which happens when $|r| < 1$. In that case, the series converges and the sum is:

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

If $|r| \geq 1$, the terms don't shrink, so the series diverges (no finite sum exists).

Example: Find the sum of the infinite series 5, 2, 0.8, 0.32, ...

1. Identify $a_1 = 5$ and $r = \frac{2}{5} = 0.4$
2. Since $|0.4| < 1$, the series converges
3. Apply the formula: $S_\infty = \frac{5}{1 - 0.4} = \frac{5}{0.6} = \frac{25}{3} \approx 8.33$

This means no matter how many terms you add, the total will never exceed $\frac{25}{3}$.

Real-World Applications

Each application below is really just the geometric sequence formula dressed up for a specific context. The key is recognizing that the "common ratio" takes a different form in each scenario.

1. Compound interest: $A = P(1 + r)^n$

   • $P$ = principal (initial investment)
   • $r$ = interest rate per compounding period (as a decimal)
   • $n$ = number of compounding periods
   • $A$ = final amount
   • Example: $1,000 at 5% annual interest for 3 years gives $A = 1000(1.05)^3 = \$1{,}157.63$

2. Population growth: $P_n = P_0(1 + r)^n$

   • $P_0$ = initial population
   • $r$ = growth rate per period (as a decimal)
   • $n$ = number of periods
   • Here the common ratio is $(1 + r)$, which is greater than 1 for a growing population

3. Depreciation: $V_n = V_0(1 - r)^n$

   • $V_0$ = initial value
   • $r$ = depreciation rate per period (as a decimal)
   • $n$ = number of periods
   • The common ratio is $(1 - r)$, which is between 0 and 1, so the value decreases over time

Related Concepts

• Arithmetic sequences have a constant difference between terms (add the same number each time), while geometric sequences have a constant ratio (multiply by the same number each time). Mixing these up is one of the most common errors on exams.
• Recursive definitions express each term using the previous term: $a_{n+1} = r \cdot a_n$. This is useful when you're building a sequence term by term rather than jumping straight to the nth term.