Ever stare at a list of number sequences and wonder which ones actually follow a pattern — and which are just messing with you? Yeah, me too. Still, it sounds like a dry math question, but once you know what to look for, it gets almost satisfying. Like spotting the one impostor in a lineup.
The short version is this: when someone asks which of the following sequences are geometric*, they're really asking whether each sequence multiplies by the same number every step. Miss that, and you'll label a perfectly arithmetic sequence as geometric — or worse, call a random list "geometric" because it looks fancy.
What Is a Geometric Sequence
Let's skip the textbook talk. That's geometric. Practically speaking, if you start at 2 and keep multiplying by 3, you get 2, 6, 18, 54. That value has a name: the common ratio*. Now, a geometric sequence is just a list of numbers where you multiply by the same value to get the next one. Easy.
But here's what most people miss. It can even be 1, which gives you a boring but totally valid sequence like 5, 5, 5, 5. All geometric. It can be negative. Still, the ratio can be a fraction. And it can be -1, which flips signs: 4, -4, 4, -4. All legal.
How the Ratio Shows Up
You find the ratio by dividing any term by the one before it. Do that for every adjacent pair. If you get the same result each time, congratulations — it's geometric. If even one pair breaks the pattern, it isn't.
Why People Confuse It With Arithmetic
Arithmetic sequences add the same number. And geometric ones multiply. That's the whole difference. Here's the thing — a sequence like 3, 7, 11, 15 adds 4 each time — not geometric, no matter how tidy it looks. I know it sounds simple, but it's easy to miss under time pressure.
Why It Matters
Why does this matter? In practice, because most people skip it and then bomb the test question or the interview problem. Knowing which of the following sequences are geometric* isn't just school trivia. It shows up in finance (compound interest is geometric), in biology (population growth), in computer science (algorithm scaling), and honestly in everyday pattern recognition.
Turns out, if you misclassify a sequence, your whole formula breaks. In practice, you'll use the wrong sum formula. You'll predict the wrong next term. You'll look at a declining balance and think it drops by a fixed amount when it's actually shrinking by a percentage — a very different beast.
And in practice, the question "which of the following sequences are geometric" is a gatekeeper. Standardized exams love it. So do coding challenges. Get comfortable with it and you stop guessing.
How It Works
Here's the thing — figuring it out is a process, not a vibe. Let's walk through how to actually do it when faced with a set of sequences.
Step 1: Write Down the Sequences
Say you're given:
- A: 2, 4, 8, 16
- B: 5, 10, 15, 20
- C: 3, -6, 12, -24
- D: 1, 1, 1, 1
- E: 2, 5, 9, 14
Don't eyeball it. Put them on paper or screen.
Step 2: Divide Consecutive Terms
For A: 4/2 = 2, 8/4 = 2, 16/8 = 2. That said, same ratio. For B: 10/5 = 2, 15/10 = 1.Geometric. Which means for C: -6/3 = -2, 12/-6 = -2, -24/12 = -2. On top of that, for E: 5/2 = 2. In real terms, same. 8. Day to day, 5, 9/5 = 1. For D: 1/1 = 1 every time. 33. Geometric (ratio 1). Not geometric. Still, not same. Geometric (yes, negative ratio counts). Because of that, nope. Think about it: 5, 20/15 ≈ 1. Not geometric.
So which of the following sequences are geometric? A, C, and D.
Step 3: Watch for Zero and Weird Cases
A sequence starting with 0 is tricky. On top of that, 0, 0, 0 is geometric (ratio undefined but consistent zeros are usually accepted). But 0, 2, 4 can't be geometric because you can't divide 2 by 0 to get a ratio. Real talk — exam writers love slipping a zero in to trip you.
Step 4: Check the Whole List, Not Just the First Two
A sequence like 2, 6, 18, 36 looks geometric at first (×3, ×3) then breaks (×2). Plus, one break disqualifies it. Always check every step.
For more on this topic, read our article on factored form of a quadratic function or check out difference between meiosis i and ii.
Step 5: Use the Formula to Confirm
The nth term of a geometric sequence is a·r^(n-1), where a is the first term and r is the ratio. Plug in. If it spits out your sequence, you're right. This is also how you catch subtle errors.
Common Mistakes
Honestly, this is the part most guides get wrong — they tell you the definition and bounce. But the mistakes are where the learning sticks.
One big one: assuming a sequence is geometric because it grows fast. Exponential growth feels geometric, but 1, 2, 4, 7, 11 grows oddly and isn't. Fast doesn't mean multiplicative.
Another: ignoring signs. Still, a sequence like 1, -2, 4, -8 is geometric with r = -2. People see the flip-flop and think "that's not a pattern." It is. Just a negative pattern.
And then there's the ratio-by-addition error. Divide, don't add. Stop. You'll see 2, 4, 8 and say "plus 2, plus 4" and get confused. The question is which of the following sequences are geometric*, not which are arithmetic.
Also, fractional ratios get skipped. 81, 27, 9, 3 has r = 1/3. That said, it's geometric. Consider this: looks like division, but division by 3 is multiplication by 1/3. Same thing.
Practical Tips
Here's what actually works when you're staring at a "which of the following sequences are geometric" problem on a timed quiz:
- Divide, don't subtract. Force yourself. Even if addition looks obvious, do the division first.
- Check the last pair. Most fakes hide at the end. First three terms lie.
- Write ratios under each jump. Like 2 →4 (×2) →8 (×2) →16 (×2). Visual beats memory.
- Expect negatives and fractions. Train your brain to accept r = -1/2 as normal.
- If zero appears, pause. Know the rules your specific context uses for zero-start sequences.
Worth knowing: some sequences are both arithmetic and geometric. Even so, only constant sequences like 4, 4, 4, 4 qualify (ratio 1, difference 0). If you spot one, it's the rare double agent.
FAQ
How do you tell if a sequence is geometric or arithmetic? Divide consecutive terms. If the quotient is constant, it's geometric. If the difference is constant instead, it's arithmetic. Geometric multiplies; arithmetic adds.
Can a geometric sequence have a negative common ratio? Yes. A negative ratio just alternates the sign of each term. Example: 3, -6, 12, -24 has r = -2 and is fully geometric.
What if the sequence starts with zero? If every term is zero, it's generally treated as geometric. If only the first term is zero and the rest aren't, you can't define a ratio, so it's not geometric in the standard sense.
Is 1, 1, 1, 1 a geometric sequence? It is. The common ratio is 1. Every term is the previous term times 1. Boring, but valid.
Why do tests ask which of the following sequences are geometric? Because it quickly checks whether you understand multiplicative patterns versus additive ones — a fundamental math skill with real-world uses
in finance, population modeling, and signal processing.
Can a geometric sequence decrease forever? Yes, as long as the absolute value of the common ratio is between 0 and 1 (exclusive), the terms shrink toward zero without ever reaching it. Here's a good example: 100, 50, 25, 12.5 continues indefinitely with r = 1/2.
What's the most common trick on these quizzes? Sequences that start geometric and break late. Something like 5, 10, 20, 40, 90 looks convincing for four terms, then betrays you. That's why checking the final pair is non-negotiable.
Conclusion
Identifying geometric sequences comes down to one disciplined habit: verify the ratio, not the vibe. Because of that, fast growth, alternating signs, and shrinking values are all fair game as long as every step multiplies by the same number. When a question asks "which of the following sequences are geometric," slow down, divide every consecutive pair, and trust the math over your first impression. Master that, and the timed quizzes stop being traps and start being free points.
