Which Of The Following Are Geometric Sequences

A geometric sequence is a fundamental concept in mathematics where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding how to identify these sequences is crucial for algebra, calculus, financial modeling, and even computer science algorithms. This guide provides a deep dive into the definition, the identification process, worked examples, and common pitfalls to help you confidently answer the question: *which of the following are geometric sequences?

The Core Definition: What Makes a Sequence Geometric?

At its heart, a geometric sequence (or geometric progression) is an ordered list of numbers where the ratio between consecutive terms remains constant. Unlike an arithmetic sequence, where you add a common difference, in a geometric sequence, you multiply by a common ratio ($r$) Easy to understand, harder to ignore. Turns out it matters..

The General Formula: The $n$-th term of a geometric sequence can be expressed as: $a_n = a_1 \cdot r^{(n-1)}$

Where:

• $a_n$ = the $n$-th term
• $a_1$ = the first term
• $r$ = the common ratio
• $n$ = the term position (1, 2, 3, ...)

The Definitive Test: To determine if a sequence ${a_1, a_2, a_3, ...}$ is geometric, calculate the ratio of successive terms: $\frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = ... = r$

If this equality holds true for all adjacent pairs, the sequence is geometric. If the ratio changes even once, it is not.

The Common Ratio ($r$): The Fingerprint of the Sequence

The common ratio dictates the behavior of the sequence. Analyzing $r$ tells you exactly how the sequence grows or decays Simple, but easy to overlook..

Critical Constraint: The common ratio cannot be zero. If $r=0$, the sequence becomes $a_1, 0, 0, 0...$, which breaks the multiplicative pattern definition after the second term (division by zero is undefined when testing $a_3/a_2$) And that's really what it comes down to. Worth knowing..

Step-by-Step Identification Strategy

When faced with a list of sequences and asked "which of the following are geometric sequences," apply this systematic checklist:

1. Check for Zero Terms: If any term is zero (and it's not the trivial all-zero sequence), the sequence is not geometric (division by zero makes the ratio undefined).
2. Calculate Ratios: Divide the 2nd term by the 1st ($r_1 = a_2/a_1$). Divide the 3rd by the 2nd ($r_2 = a_3/a_2$).
3. Compare Ratios: If $r_1 = r_2$, calculate $r_3 = a_4/a_3$ to be safe.
4. Verify Consistency: If all calculated ratios are identical, it is geometric. If they differ, it is not.
5. Beware of Patterns: Do not confuse visual patterns (like squares or cubes) with multiplicative consistency.

Worked Examples: Geometric vs. Non-Geometric

Let’s test several candidate sequences to sharpen your identification skills Easy to understand, harder to ignore. Less friction, more output..

Example A: $3, 12, 48, 192, 768 ...$

• $12 / 3 = 4$
• $48 / 12 = 4$
• $192 / 48 = 4$
• Verdict: Geometric. ($a_1=3, r=4$)

Example B: $5, 10, 20, 35, 70 ...$

• $10 / 5 = 2$
• $20 / 10 = 2$
• $35 / 20 = 1.75$ $\leftarrow$ Mismatch!
• Verdict: Not Geometric. (This looks like it almost doubles, but the pattern breaks at the 4th term).

Example C: $-2, 6, -18, 54, -162 ...$

• $6 / -2 = -3$
• $-18 / 6 = -3$
• $54 / -18 = -3$
• Verdict: Geometric. ($a_1=-2, r=-3$). The alternating signs are a hallmark of a negative ratio.

Example D: $1, 4, 9, 16, 25 ...$ (Perfect Squares)

• $4 / 1 = 4$
• $9 / 4 = 2.25$
• Verdict: Not Geometric. This is a quadratic sequence ($n^2$), not exponential.

Example E: $100, 10, 1, 0.1, 0.01 ...$

• $10 / 100 = 0.1$
• $1 / 10 = 0.1$
• $0.1 / 1 = 0.1$
• Verdict: Geometric. ($a_1=100, r=0.1$ or $1/10$). Decay sequences are geometric.

Example F: $0, 0, 0, 0 ...$

• Technically, the ratio is undefined ($0/0$). Even so, by convention, this is often treated as a degenerate geometric sequence with any $r$, or simply a constant arithmetic sequence ($d=0$). In standard multiple-choice contexts, if this appears, it is usually considered geometric (constant sequence where $r=1$ effectively).

Example G: $2, 4, 8, 16, 32, 64 ...$ vs $2, 4, 6, 8, 10 ...$

• Sequence 1: Ratios are $2, 2, 2, 2...$ $\rightarrow$ Geometric.
• Sequence 2: Ratios are $2, 1.5, 1.33...$ $\rightarrow$ Not Geometric (This is Arithmetic, $d=2$).

Common Traps and "Look-Alikes"

Standardized tests and textbooks often include sequences designed to trick you. Here are the most

...frequent offenders:

1. The "Almost Geometric" Sequence (Polynomial Growth)

Sequences like $2, 6, 12, 20, 30...$ (generated by $n^2+n$) or $1, 8, 27, 64...$ (cubes, $n^3$) often fool the eye because the terms grow rapidly.

• The Trap: Rapid growth $\neq$ Geometric.
• The Check: Calculate the ratios. For $1, 8, 27, 64$: $8/1=8$, $27/8=3.375$, $64/27 \approx 2.37$. The ratio shrinks as $n$ increases. In a true geometric sequence, the ratio is constant, regardless of how large the terms get.

2. The "Recursive Imposter" (Non-Constant Multiplier)

Consider the sequence defined by $a_1 = 2$ and $a_n = n \cdot a_{n-1}$.

• Terms: $2, 4, 12, 48, 240...$
• The Trap: Every term is the previous term multiplied by something. It feels multiplicative.
• The Check: The multiplier changes at every step ($\times 2, \times 3, \times 4, \times 5$). Because the ratio depends on $n$, it is not geometric. (This is actually the factorial sequence scaled by 2).

3. The Alternating Sign Trap (Non-Geometric Oscillation)

Sequence: $1, -2, 3, -4, 5, -6...$

• The Trap: The signs alternate, just like Example C ($-2, 6, -18...$).
• The Check: Ratios: $-2/1 = -2$; $3/-2 = -1.5$; $-4/3 \approx -1.33$. The magnitude of the ratio changes. True geometric alternation requires a constant negative ratio (e.g., $r=-2$ yields $1, -2, 4, -8...$).

4. The "Hidden Zero" Term

Sequence: $5, 0, 0, 0, 0...$

• The Trap: It looks like it "dies out" to zero, similar to a decay sequence ($r < 1$).
• The Check: $a_2/a_1 = 0/5 = 0$. $a_3/a_2 = 0/0$ (Undefined).
• Verdict: Not Geometric. A single zero term (unless it is the trivial all-zero sequence) breaks the definition because the ratio $a_{n+1}/a_n$ becomes undefined at the transition out of zero, or forces $r=0$ which forces all subsequent terms to be zero—but the ratio from zero remains undefined. Standard convention excludes these.

5. Fractions and Radicals: The "Messy Ratio" Illusion

Sequence: $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}...$

• The Trap: The denominators grow ($2, 6, 18, 54$), looking like a pattern, but the fractions themselves look "complicated."
• The Check: Divide term 2 by term 1: $\frac{1/6}{1/2} = \frac{1}{6} \times \frac{2}{1} = \frac{1}{3}$. Term 3 / Term 2: $\frac{1/18}{1/6} = \frac{1}{3}$.
• Verdict: Geometric. ($r = 1/3$). Never let ugly fractions scare you; the division rule works exactly the same way.

Summary: The "Litmus Test" Protocol

When you encounter a sequence on an exam or in a problem set, do not guess. Execute this 15-second protocol:

1. Scan for Zeros. If a non-initial term is zero $\rightarrow$ Not Geometric (usually).
2. Pick Two Gaps. Calculate $a_2/a_1$ and $a_3/a_2$.
3. Compare. If they differ $\rightarrow$ Not Geometric. Stop.
4. Verify (Optional but Recommended). Calculate $a_4/a_3$. If it matches $\rightarrow$ Geometric.
5. State $a_1$ and $r$. Explicitly write them down. This prevents sign errors (e.g., confusing $r=-3$ with $r=3$) and prepares you for the $n$-th term formula $a_n = a_1 r^{n-1}$ or the sum formula $S

...the sum formula $S_n = \frac{a_1(1 - r^n)}{1 - r}$ (for $r \neq 1$). With $a_1$ and $r$ identified, you can jump directly to any term or partial sum without listing the entire sequence Small thing, real impact. No workaround needed..

Conclusion

The secret to geometric sequences is deceptively simple: ignore the surface pattern and hunt for the constant ratio. A sequence can alternate signs, collapse into fractions, or look chaotic, yet still be geometric if every step multiplies by the same fixed number. Conversely, a sequence that looks orderly—like consecutive integers or squares—can fail the test the moment the ratio shifts.

By applying the litmus test—screening for zeros, comparing consecutive quotients, and verifying with a third gap—you protect yourself against the most common traps. Once verified, the explicit formula $a_n = a_1 r^{n-1}$ and the sum formula become reliable tools rather than memorized tricks That's the whole idea..

When in doubt, divide. Now, if the quotient holds, it’s geometric. If not, look elsewhere. That single habit will save you from costly errors on exams and in real-world models of growth and decay Worth keeping that in mind..