Geometric Sequences worksheet for practicing identification, common ratio calculation, and sequence completion.

Let’s go step by step through each section of the worksheet. We’ll solve every problem carefully and check our work.

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Section A: Circle all the geometric sequences below.

A geometric sequence is one where you multiply (or divide) by the same number each time to get the next term. That number is called the “common ratio”.

Let’s test each sequence:

1. 1, 1, 2, 3, 5, 8, ... → This is Fibonacci — add previous two terms. Not geometric. ✘
2. 6000, 3000, 1500, ... → Divide by 2 each time? 6000 ÷ 2 = 3000, 3000 ÷ 2 = 1500 → YES ✔
3. 1, 3, 6, 10, 15, ... → Differences increase: +2, +3, +4... not multiplying → ✘
4. 1, 1/3, 1/4, 1/8, ... → From 1 to 1/3: ×1/3; 1/3 to 1/4: ×3/4? Not same → ✘
5. -8, -16, -32, -64, ... → Multiply by 2 each time? -8×2=-16, -16×2=-32 → YES ✔
6. x, x+1, x+2, x+3, ... → Adding 1 each time → arithmetic, not geometric → ✘
7. 10, 100, 1000, 10000, ... → ×10 each time → YES ✔
8. -1, 1, -1, 1, -1, ... → ×(-1) each time → YES ✔
9. 4, 6, 9, 13.5, ... → 4×1.5=6, 6×1.5=9, 9×1.5=13.5 → YES ✔
10. 5, 10, 15, 20, ... → Add 5 → arithmetic → ✘
11. 0.1, 0.2, 0.3, 0.4, ... → Add 0.1 → arithmetic → ✘
12. a, 2a, 4a, 8a, ... → ×2 each time → YES ✔

✔ So the geometric sequences are:
- 6000, 3000, 1500, ...
- -8, -16, -32, -64, ...
- 10, 100, 1000, 10000, ...
- -1, 1, -1, 1, -1, ...
- 4, 6, 9, 13.5, ...
- a, 2a, 4a, 8a, ...

Now finish the sentence:

> A geometric series is the sum of the terms in a geometric sequence.

*(Note: The question says “series”, which means adding them up. But sometimes people say “sequence” when they mean “series”. Since it says “series”, we give the correct definition.)*

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Section B: Find the common ratio of the geometric sequences.

Common ratio = second term ÷ first term (or any term ÷ previous term)

1) 5, 20, 80, 320 → 20÷5 = 4
2) 1, -5, 25, -125 → -5÷1 = -5
3) 3, 4.5, 6.75 → 4.5÷3 = 1.5
4) 3.2, 6.4, 12.8 → 6.4÷3.2 = 2
5) 6000, 600, 60 → 600÷6000 = 0.1
6) 1, ?, 9, ?, 81 → Let’s find ratio: from 1 to 9 is two steps → so r² = 9 → r = 3 or -3. But since 81 is positive, likely r=3. Check: 1, 3, 9, 27, 81 → yes → 3
7) 1, 1/3, 1/9, 1/27 → (1/3)÷1 = 1/3
8) 10, 2, 0.4, 0.125 → 2÷10 = 0.2 (check: 2×0.2=0.4, 0.4×0.2=0.08? Wait — 0.4×0.2=0.08 but given is 0.125? Hmm… let’s recalculate: 10 → 2 → 0.4 → 0.125? 2÷10=0.2, 0.4÷2=0.2, 0.125÷0.4=0.3125? That doesn’t match. Wait — maybe typo? Or did I misread? Let me check again: 10, 2, 0.4, 0.125 — actually 0.4 × 0.3125 = 0.125? No, that’s not consistent. Wait — perhaps it’s 10, 2, 0.4, 0.08? But it says 0.125. Let me compute ratios: 2/10=0.2, 0.4/2=0.2, then 0.125/0.4=0.3125 — inconsistency. Maybe it’s supposed to be 0.08? But as written, it’s not geometric? Wait — perhaps I made a mistake. Let me try dividing backwards: 0.125 ÷ 0.4 = 5/16 = 0.3125, not 0.2. So this might be an error? But looking at pattern: 10, 2, 0.4 — that’s ÷5 each time? 10÷5=2, 2÷5=0.4, 0.4÷5=0.08 — but it says 0.125. Hmm. Alternatively, maybe it’s ×0.2? 10×0.2=2, 2×0.2=0.4, 0.4×0.2=0.08 — still not 0.125. Perhaps it’s a different ratio? Let’s see: if 10 to 2 is ×0.2, 2 to 0.4 is ×0.2, then 0.4 to ? should be ×0.2 = 0.08, but it’s 0.125. So unless there’s a typo, this isn’t geometric. But wait — maybe I miscalculated 0.4 × what = 0.125? 0.125 / 0.4 = 125/400 = 5/16 = 0.3125. Not matching. Perhaps the sequence is 10, 2, 0.4, 0.08? But it says 0.125. Let me check original image description — user wrote: "10, 2, 0.4, 0.125, ..." — oh! Wait — 0.125 is 1/8. And 0.4 is 2/5. Not obvious. Alternatively, maybe it's 10, 2, 0.4, 0.08 — but written wrong? Or perhaps it's 10, 2, 0.4, 0.125 with ratio changing? That can't be. Another idea: maybe it's 10, 2, 0.4, 0.125 — let's do exact fractions: 10, 2, 2/5, 1/8. Ratios: 2/10=1/5, (2/5)/2=1/5, (1/8)/(2/5)= (1/8)*(5/2)=5/16 — not 1/5. So not geometric. But the problem says "find the common ratio", implying it is geometric. Perhaps it's a typo and should be 0.08? In many worksheets, 10, 2, 0.4, 0.08 is standard. Given that, and since 0.4 * 0.2 = 0.08, I think it's meant to be 0.08. But user wrote 0.125. Let me double-check the user input: "8) 10, 2, 0.4, 0.125, ..." — oh! Wait — 0.125 is 1/8, and 0.4 is 2/5. Perhaps it's 10, 2, 0.4, 0.08 — but written as 0.125 by mistake? Or maybe it's 10, 2, 0.4, 0.125 with ratio 0.2 for first three, then changes? That doesn't make sense. Another possibility: maybe it's 10, 2, 0.4, 0.125 — and we need to find ratio between consecutive terms: 2/10=0.2, 0.4/2=0.2, 0.125/0.4=0.3125 — not constant. So either it's not geometric, or there's a typo. But since the section says "geometric sequences", I'll assume it's intended to be geometric, and perhaps 0.125 is a mistake for 0.08. In that case, ratio is 0.2. Or maybe it's 1/5. Let's look at other problems. Problem 9 is x,x^2,x^3 — ratio x. Problem 10 is clear. For now, I'll go with 0.2, assuming 0.125 is a typo for 0.08. Because 10 * 0.2 = 2, 2 * 0.2 = 0.4, 0.4 * 0.2 = 0.08. So probably meant 0.08. I'll use 0.2.

But let's confirm with calculation: if it were 0.125, then from 0.4 to 0.125 is multiply by 5/16, which is not nice. Whereas 0.2 is clean. So I'll proceed with 0.2.

9) x, x², x³, x⁴ → ratio = x²/x = x
10) -7, -14, -28, -56 → -14/-7 = 2

So answers for Section B:

1) 4
2) -5
3) 1.5
4) 2
5) 0.1
6) 3
7) 1/3
8) 0.2 (assuming typo)
9) x
10) 2

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Section C: Fill the gaps in these geometric sequences.

We need to find missing terms using the common ratio.

1) 2, __, 200, __, 20000
From 2 to 200: that’s two steps. So 2 * r² = 200 → r² = 100 → r=10 or r=-10. Then next term after 200 is 200*r, then 200*r²=20000. If r=10: 2, 20, 200, 2000, 20000 → fits. If r=-10: 2, -20, 200, -2000, 20000 — also fits, but usually we take positive unless specified. But both are valid. However, since no negative signs, probably r=10. So gaps: 20 and 2000

2) __, 15, 75, __
From 15 to 75: ×5. So previous term: 15÷5=3. Next term: 75×5=375. So 3 and 375

3) 1, 4, __, __
Ratio: 4/1=4. So next: 4×4=16, then 16×4=64. So 16, 64

4) 7, __, __, 189
Three steps from 7 to 189. So 7 * r³ = 189 → r³ = 27 → r=3. So terms: 7, 21, 63, 189. Gaps: 21, 63

5) 200, __, 50, __
From 200 to 50: two steps. 200 * r² = 50 → r² = 50/200 = 1/4 → r=1/2 or -1/2. If r=1/2: 200, 100, 50, 25. If r=-1/2: 200, -100, 50, -25. Both possible, but likely positive. So 100, 25

6) __, 12, -36, __
From 12 to -36: ×(-3). So previous: 12 ÷ (-3) = -4. Next: -36 × (-3) = 108. So -4, 108

7) 8, __, 8, __
If it goes 8, ?, 8, ? — and geometric, then ratio must be such that 8 * r² = 8 → r²=1 → r=1 or r=-1. If r=1: 8,8,8,8. If r=-1: 8,-8,8,-8. Both work. But since it has blanks, probably r=-1. So -8, -8? Wait: positions: term1=8, term2=?, term3=8, term4=? So if r=-1: term2=8*(-1)=-8, term3=-8*(-1)=8, term4=8*(-1)=-8. So gaps: -8, -8

8) 1/3, __, 1/12, __
From 1/3 to 1/12: two steps. (1/3)*r² = 1/12 → r² = (1/12)/(1/3) = 3/12 = 1/4 → r=1/2 or -1/2. If r=1/2: 1/3, 1/6, 1/12, 1/24. If r=-1/2: 1/3, -1/6, 1/12, -1/24. Likely positive. So 1/6, 1/24

9) 4096, 512, __, 8, __
From 4096 to 512: 512/4096 = 1/8. So ratio r=1/8? Then next: 512*(1/8)=64, then 64*(1/8)=8, then 8*(1/8)=1. So gaps: 64, 1

10) -20, -100, __, __
Ratio: -100 / -20 = 5. So next: -100*5 = -500, then -500*5 = -2500. So -500, -2500

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Section D: Show me...

1) A sequence with common ratio 6: Start with any number, say 1: 1, 6, 36, 216, ...
2) A decreasing geometric sequence: Ratio between 0 and 1, e.g., 100, 50, 25, 12.5, ... (ratio 0.5)
3) A sequence with common ratio -2: Start with 1: 1, -2, 4, -8, ...

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Final Answer:

Section A Geometric Sequences:
- 6000, 3000, 1500, ...
- -8, -16, -32, -64, ...
- 10, 100, 1000, 10000, ...
- -1, 1, -1, 1, -1, ...
- 4, 6, 9, 13.5, ...
- a, 2a, 4a, 8a, ...

Sentence: A geometric series is the sum of the terms in a geometric sequence.

Section B Common Ratios:
1) 4
2) -5
3) 1.5
4) 2
5) 0.1
6) 3
7) 1/3
8) 0.2
9) x
10) 2

Section C Filled Gaps:
1) 20, 2000
2) 3, 375
3) 16, 64
4) 21, 63
5) 100, 25
6) -4, 108
7) -8, -8
8) 1/6, 1/24
9) 64, 1
10) -500, -2500

Section D Examples:
1) 1, 6, 36, 216, ...
2) 100, 50, 25, 12.5, ...
3) 1, -2, 4, -8, ...