Spotting geometric sequences with the 2, 4, 8, 16 pattern shows a constant ratio.

Geometric sequences are like secret multipliers playing a steady tune as you move from term to term. If you’ve ever watched something grow by a fixed factor—like bacteria doubling, or a savings account that compounds—then you already have a feel for this idea. Let’s break it down in a way that fits right into how you’d approach the HSPT math topics, with a little math detective work you can reuse again and again.

What exactly is a geometric sequence?

• Here’s the core idea: in a geometric sequence, every term is obtained by multiplying the previous term by a fixed number, called the common ratio. If that ratio is r, then a, ar, ar^2, ar^3, and so on are the terms of the sequence.

• A quick mental image: think of a plant that doubles in size every week. If it starts at 2 inches, after one week it’s 4 inches, after two weeks 8 inches, then 16 inches, and so on. The number you multiply by each time is the common ratio.

Let’s test the given options like a careful math sleuth

The question gives four sequences and asks which one is geometric. The key move is to look for a constant ratio between consecutive terms.

• Option A: 1, 2, 3, 4

• Check the ratios: 2 ÷ 1 = 2, 3 ÷ 2 = 1.5, 4 ÷ 3 ≈ 1.33. Those ratios aren’t the same, so this one isn’t geometric. It’s actually a simple arithmetic feel—each step adds the same amount (a difference of 1), not a constant multiplier.

• Option B: 2, 4, 8, 16

• Ratios: 4 ÷ 2 = 2, 8 ÷ 4 = 2, 16 ÷ 8 = 2. Here the ratio is consistently 2. That’s the hallmark of a geometric sequence. This is the one that fits the definition.

• Option C: 5, 10, 15, 20

• Ratios: 10 ÷ 5 = 2, 15 ÷ 10 = 1.5, 20 ÷ 15 ≈ 1.33. Not constant. This is another arithmetic feel—the differences are all 5, not a fixed multiplier.

• Option D: 10, 20, 30, 40

• Ratios: 20 ÷ 10 = 2, 30 ÷ 20 = 1.5, 40 ÷ 30 ≈ 1.33. Same story as C—no fixed ratio, so not geometric.

The correct answer is B, 2, 4, 8, 16. The number you multiply by each time—the common ratio—is 2, and it stays the same all along the line of terms.

Why this matters beyond a single question

You don’t need to memorize a long list of sequences to spot a geometric one. The trick is to ask: “Is there a consistent multiplier from one term to the next?” If yes, you’re in geometric territory. If the change from term to term is a constant addition, you’re in arithmetic territory instead. It’s a simple fork in the road, but one that makes a big difference when you’re quickly evaluating sequences.

Let’s contrast with arithmetic sequences for a moment

• Arching through this topic is a small but important distinction. In an arithmetic sequence, you add the same number every time. That “same-number-addition” is called the common difference.

• A quick look at the wrong options helps the mind lock onto the pattern:

• Option A grows by adding 1 each step (1, 2, 3, 4). Here the difference is 1, not a multiplier.

• Option C grows by adding 5 each step (5, 10, 15, 20). The difference is 5.

• Option D grows by adding 10 each step (10, 20, 30, 40). The difference is 10.

• In short: arithmetic sequences march forward by addition, geometric sequences march forward by multiplication.

A mental model you can carry around

Think of a geometric sequence as a staircase where each step is a fixed factor taller than the last. If you know the bottom step and the height of each step in multiplicative terms, you can predict a lot just by multiplying. It’s not about guessing; it’s about recognizing a steady multiplier at work.

A little practice you can try on your own

• Suppose you’re given 3, 9, 27, 81. Do you see the pattern? Ratios are 9 ÷ 3 = 3, 27 ÷ 9 = 3, 81 ÷ 27 = 3. Constant ratio of 3. This is geometric.

• What about 6, 12, 24, 48? Ratios are 12 ÷ 6 = 2, 24 ÷ 12 = 2, 48 ÷ 24 = 2. Again, geometric with ratio 2.

• If you spot two consecutive ratios that match, you’re close to declaring a sequence geometric. If one ratio breaks the pattern, the sequence isn’t geometric.

A quick note on edge cases

• Some sequences flip signs or hit zero. If you multiply by zero, all following terms will be zero. If the ratio isn’t consistently defined (for example, you have a zero in the middle where you can’t divide by zero), you need to be careful. The clean, textbook definition expects a nonzero common ratio when you compare consecutive terms.

• Negative ratios are possible too. If you had 5, -10, 20, -40, the ratio would oscillate in sign but still be consistent in magnitude (a ratio of -2). That’s still geometric, though it can feel a bit odd at first glance.

A practical takeaway you can carry forward

• When you’re faced with a sequence, a fast check is to pick a couple of consecutive terms and compute the ratio. If the same ratio pops up again and again, you’ve got a geometric sequence.

• If you only see one ratio and then a different one, or if some steps don’t fit, you’re looking at something else—most likely arithmetic or a nonstandard pattern.

A tiny note about tools and visualization

If you’re ever unsure, a quick sketch can help. Plot the terms on graph paper or use a tool like Desmos to see how the numbers behave. A geometric sequence will look like a staircase that jumps more and more quickly (in a multiplicative sense) as you move along, whereas an arithmetic sequence climbs in straight, even steps.

The big idea in one sentence

A geometric sequence is built by multiplying the previous term by a fixed ratio every time. If that ratio stays the same, you’re in classic geometric territory. If not, you’re probably looking at an arithmetic sequence or a different kind of pattern.

A gentle closing thought

Patterns like these aren’t just about ticking off boxes on a worksheet. They’re ways your brain learns to recognize structure—how numbers like to cooperate, how multiplication creates momentum, and how a single, steady rule can govern a whole line of terms. When you’re comfortable with the concept, you’ll see it in a lot of places, from science to everyday money matters, sometimes even in the way a playlist grows or a crowd of fans builds a chorus.

If you want a quick recap, remember this: geometric = constant multiplier from one term to the next. Constant ratio, always. And the sequence that fits that bill among the options here is 2, 4, 8, 16. The rest march by addition, not multiplication. With that lens, you’ll be able to spot the pattern in a glance and move on with confidence.