Recognize Geometric Sequences by a Constant Ratio
Geometric sequence
A sequence in which the ratio between consecutive terms is constant (each term is found by multiplying the previous term by the same number).
- A sequence is an ordered list of numbers (called terms) such as $6, 42, 294, \ldots$.
- In many practical settings, the terms change by the same multiplier each step, rather than by the same amount.
- That is exactly the idea of a geometric sequence.
- A sequence $u_n$ is geometric if there is a constant number $r$ such that $$\frac{u_{n+1}}{u_n}=r$$ for every term where this division is defined.
- Different books use different letters.
- You may see the first term written as $a$ and the ratio as $q$.
- We use $u_1$ for the first term and $r$ for the common ratio.
Understand the Common Ratio and How It Controls Behavior
Common ratio
The constant multiplier $r$ in a geometric sequence, found by dividing a term by the previous term: $r=\frac{u_{n+1}}{u_n}$.
If a geometric sequence begins $4, 12, 36, 108, \ldots$, then $$r=\frac{12}{4}=\frac{36}{12}=\frac{108}{36}=3,$$ so every term is the previous term multiplied by 3.
Different Values of $r$ Produce Different Patterns
The value of the common ratio $r$ determines the long-term behavior.
- $r>1$: if $u_1>0$, the terms increase rapidly (geometric growth).
- $0<r<1$: if $u_1>0$, the terms decrease toward 0 (geometric decay).
- $r=1$: all terms are equal (constant sequence).
- $r<0$: the terms alternate in sign, because multiplying by a negative number flips the sign each step.
- The sequence $5, -10, 20, -40, 80, \ldots$ is geometric with $$r=\frac{-10}{5}=-2$$
- Because $r$ is negative, the signs alternate: $+, -, +, -, +, \ldots$.
- When $r$ is negative, the terms of a geometric sequence alternate in sign.
- The sizes still grow if $|r|>1$.
- The constant sequence $3, 3, 3, 3, \ldots$ is geometric (with $r=1$).
- However, the all-zero sequence $0, 0, 0, 0, \ldots$ is usually not treated as geometric because ratios like $\frac{0}{0}$ are undefined, so the "constant ratio" test breaks down.
Use Recursive and Explicit Formulae Correctly
Two types of rules are used to describe sequences.
Recursive Form for a Geometric Sequence
If the first term is $u_1$ and the common ratio is $r$, the recursive definition is $$u_{n+1}=r u_n,\quad u_1\text{ given}$$
This matches the idea "multiply by $r$ to get the next term."
Explicit Form for a Geometric Sequence
- Starting from $u_1$:
- $u_2=u_1r$
- $u_3=u_1r^2$
- $u_4=u_1r^3$
- So the general (explicit) term is $$u_n=u_1r^{n-1}$$
- To check the power $n-1$, substitute $n=1$: you should get $u_1$.
- Indeed $u_1r^0=u_1$.
Find Missing Information by Forming Equations
- Many questions give partial information (for example, two terms, or a relationship between terms) and ask you to determine $u_1$, $r$, or a later term.
- The explicit formula is especially powerful because it turns patterns into algebra.
- A geometric sequence begins $6, 42, 294, \ldots$.
- Find $r$: $$r=\frac{42}{6}=7$$
- Find $u_5$: $$u_n=6\cdot 7^{n-1} \Rightarrow u_5=6\cdot 7^4=6\cdot 2\,401=14\,406$$
A geometric sequence has common ratio $4$. The second term is 9 more than the first term. Find $u_1$.
Solution
Write the second term in two ways:
- from the ratio: $u_2=4u_1$
- from the "9 more" information: $u_2=u_1+9$
Equate them: $4u_1=u_1+9 \Rightarrow 3u_1=9 \Rightarrow u_1=3$.
If a question says a term is "more than" or "less than" another term, write two expressions for the same term and set them equal.
Solve "When Does It Exceed…?" Problems with Inequalities and Logs
Take note that logarithms are covered in the next topic, so don't worry if the following material seems complicated.
- Geometric growth can become very large, very quickly.
- A common type of task is: "find the first term that exceeds a certain value."
- For $u_n=6\cdot 7^{n-1}$, find the first term to exceed one million.
- Solve the inequality: $$6\cdot 7^{n-1}>1\,000\,000$$
- Divide by 6: $$7^{n-1}>\frac{1\,000\,000}{6}\approx 166\,666.7$$
- Take logarithms: $$(n-1)\log 7>\log(166\,666.7)$$
- So $$n-1>\frac{\log(166\,666.7)}{\log 7}\approx 6.18,$$ hence $n>7.18$.
- The first integer value that works is $n=8$.
- Therefore, the first term to exceed one million is $$u_8=6\cdot 7^{7}=4\,941\,258$$
In inequalities like $u_1r^{n-1}>k$, isolate the exponential part first, then use logs, then round $n$ up to the next integer (because $n$ is a term number).
Decide Whether a Sequence Is Geometric (or Not)
To test whether a sequence is geometric, check whether the ratios of consecutive terms are constant.
A reliable method:
- Compute $\frac{u_2}{u_1}$ and $\frac{u_3}{u_2}$.
- If these ratios are equal (and division is defined), the sequence is geometric.
Let's classify each of the following sequences:
- $18, 9, 4.5, 2.25, \ldots$
- $\frac{9}{18}=\frac{4.5}{9}=\frac{2.25}{4.5}=0.5$
- So it is geometric with $r=0.5$.
- $ 1, 4, 9, 16, 25, \ldots$
- Ratios are $4, \frac{9}{4}, \frac{16}{9}, \ldots$ (not constant), so it is not geometric.
- $13, 39, 117, 351, \ldots$
- $\frac{39}{13}=\frac{117}{39}=\frac{351}{117}=3$
- So it is geometric with $r=3$.
- Do not confuse "constant difference" (arithmetic) with "constant ratio" (geometric).
- For $4, 7, 10, 13, \ldots$ the differences are constant (3), but the ratios are not.
Link Geometric Sequences to Real Contexts
Geometric sequences model situations where the same percentage change happens repeatedly.
- Compound interest: multiply by $(1+i)$ each time period.
- Population growth at a fixed rate: multiply by $(1+g)$ each period.
- Depreciation/decay at a fixed percentage: multiply by $(1-d)$ each period.
- An investment of \$150 at 5% compound interest per year follows a geometric sequence with $u_1=150$ and $r=1.05$.
- After 3 years, you have applied the multiplier $1.05$ three times: $$u_4=150\cdot 1.05^{3}\approx 173.64$$
- Here term 1 is the initial amount (year 0), then terms 2, 3, 4 are after 1, 2, 3 years.
- A sequence has $u_1=12$ and $u_3=48$. Find the possible values of $r$.
- For $u_1=5$ and $r=-\frac{1}{2}$, write the first four terms.
- Is $0, 0, 0, 0, \ldots$ geometric? Explain using the ratio idea.
