Geometric Sequences Worksheet | Printable PDF Worksheets - Free Printable

Let's solve this Geometric Sequences worksheet step by step, explaining each section clearly.

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🔷 Section A: Circle all the geometric sequences below

A geometric sequence has a common ratio — each term is found by multiplying the previous term by a constant number (the common ratio).

We’ll check each sequence:

1. 1, 1, 2, 3, 5, 8, ...
→ Not geometric. This is the Fibonacci sequence (additive), not multiplicative. ✘

2. 6000, 3000, 1500, ...
→ 3000 ÷ 6000 = 0.5, 1500 ÷ 3000 = 0.5 → Common ratio = 0.5 ✔

3. 1, 3, 6, 10, 15, ...
→ Differences: +2, +3, +4, +5 → Arithmetic? No, not geometric. ✘

4. 1, 1/3, 1/4, 1/8, ...
→ 1/3 ÷ 1 = 1/3, 1/4 ÷ 1/3 = 3/4 ≠ 1/3 → Not consistent → ✘

5. -8, -16, -32, -64, ...
→ -16 ÷ -8 = 2, -32 ÷ -16 = 2 → Common ratio = 2 ✔

6. x, x+1, x+2, x+3, ...
→ Adds 1 each time → Arithmetic, not geometric ✘

7. 10, 100, 1000, 10000, ...
→ ×10 each time → Ratio = 10 ✔

8. -1, 1, -1, 1, -1, ...
→ Alternating signs: ×(-1) each time → Ratio = -1 ✔

9. 4, 6, 9, 13.5, ...
→ 6 ÷ 4 = 1.5, 9 ÷ 6 = 1.5, 13.5 ÷ 9 = 1.5 → Ratio = 1.5 ✔

10. 5, 10, 15, 20, ...
→ Add 5 each time → Arithmetic ✘

11. 0.1, 0.2, 0.3, 0.4, ...
→ Add 0.1 → Arithmetic ✘

12. a, 2a, 4a, 8a, ...
→ ×2 each time → Ratio = 2 ✔

✔ Geometric sequences to circle:
- 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, ...

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📝 Finish the sentence:

> A geometric series is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio.

✔ Answer:
A geometric series is a sequence where each term is obtained by multiplying the previous term by a constant (common ratio).

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

Use: $ r = \frac{\text{next term}}{\text{previous term}} $

1. 5, 20, 80, 320, ...
→ 20 ÷ 5 = 4 → ✔ 4

2. 1, -5, 25, -125, 625, ...
→ -5 ÷ 1 = -5 → ✔ -5

3. 3, 4.5, 6.75, 10.125, ...
→ 4.5 ÷ 3 = 1.5 → ✔ 1.5

4. 3.2, 6.4, 12.8, 25.6, ...
→ 6.4 ÷ 3.2 = 2 → ✔ 2

5. 6000, 600, 60, 6, ...
→ 600 ÷ 6000 = 0.1 → ✔ 0.1

6. 1, ?, 9, ?, 81, ...
→ We know it’s geometric. Let’s find the pattern:
From 1 to 9 in two steps → So $ r^2 = 9 $ → $ r = 3 $ or $ r = -3 $
But since 1 → ? → 9 → ..., and 81 comes later, let's test:
If $ r = 3 $: 1, 3, 9, 27, 81 → fits!
So common ratio = ✔ 3

7. 1, 1/3, 1/9, 1/27, ...
→ (1/3) ÷ 1 = 1/3 → ✔ 1/3

8. 10, 2, 0.4, 0.125, ...
→ 2 ÷ 10 = 0.2, 0.4 ÷ 2 = 0.2, 0.125 ÷ 0.4 = 0.3125 → Wait, not equal?
Check: 0.4 / 2 = 0.2, but 0.125 / 0.4 = 0.3125 → Not consistent.
Wait — typo? Let's see:
10 → 2: divide by 5
2 → 0.4: divide by 5
0.4 → 0.08 would be next → But it says 0.125 → ✘ Not geometric?

Wait — maybe error in sequence?
Actually:
10 → 2 → 0.4 → 0.08 → but given as 0.125 → doesn't fit.
But 0.125 = 1/8, 0.4 = 2/5 → not matching.

Let’s recheck:
2 / 10 = 0.2
0.4 / 2 = 0.2
0.125 / 0.4 = 0.3125 ≠ 0.2 → Not geometric?
But the question assumes it is. Maybe typo?

Wait — perhaps it's 10, 2, 0.4, 0.08, ...?
But as written: 10, 2, 0.4, 0.125, ... → inconsistent.

Let's assume it's supposed to be ×0.2:
10 → 2 (×0.2), 2 → 0.4 (×0.2), 0.4 → 0.08 (×0.2), but 0.125 ≠ 0.08 → so likely typo.

However, if we go with what's written:
0.125 / 0.4 = 0.3125 → not same → not geometric.

But wait — could it be 10, 2, 0.4, 0.08, ...? Then ratio = 0.2.

But as printed: 0.125 → probably mistake.

Alternatively, maybe it's meant to be 10, 2, 0.4, 0.08, but it says 0.125.

Let’s skip for now — perhaps it's intended to be ×0.2, so r = 0.2

✔ Answer: 0.2 (assuming typo)

Or double-check:
10 → 2 → ×0.2
2 → 0.4 → ×0.2
0.4 → 0.08 → should be next
But it says 0.125 → that's wrong.

So unless there's a different logic, this sequence is not geometric. But since it's listed, likely intended ratio is 0.2

✔ 0.2

9. x, x², x³, x⁴, ...
→ Each term is multiplied by x → ✔ x

10. -7, -14, -28, -56, -112, ...
→ -14 ÷ -7 = 2, -28 ÷ -14 = 2 → ✔ 2

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✔ Section B Answers:

| # | Sequence | Common Ratio |
|---|---------|--------------|
| 1 | 5, 20, 80, ... | 4 |
| 2 | 1, -5, 25, ... | -5 |
| 3 | 3, 4.5, 6.75, ... | 1.5 |
| 4 | 3.2, 6.4, ... | 2 |
| 5 | 6000, 600, ... | 0.1 |
| 6 | 1, ?, 9, ?, 81 | 3 |
| 7 | 1, 1/3, 1/9, ... | 1/3 |
| 8 | 10, 2, 0.4, 0.125 | 0.2 (assume typo) |
| 9 | x, x², x³, ... | x |
|10 | -7, -14, -28, ... | 2 |

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🔷 Section C: Fill the gaps in these geometric sequences

We use: $ a_n = a_{n-1} \times r $

#### 1) 2, [ ], 200, [ ], 20000, ...
- First: 2 → ? → 200
- Let ratio = r
- 2 × r² = 200 → r² = 100 → r = 10 (since positive)
- So:
2 → 2×10 = 20 → 20×10 = 200 → 200×10 = 2000 → 2000×10 = 20000
- ✔ Gaps: 20, 2000

#### 2) [ ], 15, 75, [ ], ...
- 15 → 75 → ratio = 75 ÷ 15 = 5
- So previous term: 15 ÷ 5 = 3
- Next term: 75 × 5 = 375
- ✔ Gaps: 3, 375

#### 3) 1, 4, [ ], [ ], ...
- Ratio = 4 ÷ 1 = 4
- Next: 4 × 4 = 16, then 16 × 4 = 64
- ✔ Gaps: 16, 64

#### 4) 7, [ ], [ ], 189, ...
- Let ratio = r
- 7 → ? → ? → 189 → so $ 7 \times r^3 = 189 $
- $ r^3 = 189 ÷ 7 = 27 $ → $ r = 3 $
- So:
7 → 7×3 = 21 → 21×3 = 63 → 63×3 = 189
- ✔ Gaps: 21, 63

#### 5) 200, [ ], 50, [ ], ...
- 200 → ? → 50
- So $ 200 \times r^2 = 50 $ → $ r^2 = 50/200 = 0.25 $ → $ r = 0.5 $
- So:
200 → 200×0.5 = 100 → 100×0.5 = 50 → 50×0.5 = 25
- ✔ Gaps: 100, 25

#### 6) [ ], 12, -36, [ ], ...
- 12 → -36 → ratio = -36 ÷ 12 = -3
- Previous: 12 ÷ (-3) = -4
- Next: -36 × (-3) = 108
- ✔ Gaps: -4, 108

#### 7) 8, [ ], 8, [ ], ...
- 8 → ? → 8 → so it goes back to 8
- So ratio must be such that $ 8 \times r \times r = 8 $ → $ r^2 = 1 $ → $ r = 1 $ or $ r = -1 $
- Try $ r = -1 $:
8 → -8 → 8 → -8 → ...
- So:
8 → -8 → 8 → -8
- ✔ Gaps: -8, -8

#### 8) 1/3, [ ], 1/12, [ ], ...
- Let ratio = r
- $ \frac{1}{3} \times r^2 = \frac{1}{12} $
- $ r^2 = \frac{1}{12} \div \frac{1}{3} = \frac{1}{12} \times 3 = \frac{1}{4} $
- $ r = \pm \frac{1}{2} $
- Try $ r = \frac{1}{2} $:
$ \frac{1}{3} \to \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \to \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} $
- Then next: $ \frac{1}{12} \times \frac{1}{2} = \frac{1}{24} $
- ✔ Gaps: 1/6, 1/24

#### 9) 4096, 512, [ ], 8, [ ], ...
- 512 ÷ 4096 = 0.125 = 1/8 → so ratio = 1/8
- 512 × 1/8 = 64
- 8 × 1/8 = 1
- So:
4096 → 512 → 64 → 8 → 1
- ✔ Gaps: 64, 1

#### 10) -20, -100, [ ], [ ], ...
- -100 ÷ -20 = 5 → ratio = 5
- Next: -100 × 5 = -500
- Then: -500 × 5 = -2500
- ✔ Gaps: -500, -2500

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✔ Section C Answers:

| # | Sequence | Gaps |
|---|--------|------|
| 1 | 2, _, 200, _, 20000 | 20, 2000 |
| 2 | _, 15, 75, _ | 3, 375 |
| 3 | 1, 4, _, _ | 16, 64 |
| 4 | 7, _, _, 189 | 21, 63 |
| 5 | 200, _, 50, _ | 100, 25 |
| 6 | _, 12, -36, _ | -4, 108 |
| 7 | 8, _, 8, _ | -8, -8 |
| 8 | 1/3, _, 1/12, _ | 1/6, 1/24 |
| 9 | 4096, 512, _, 8, _ | 64, 1 |
|10 | -20, -100, _, _ | -500, -2500 |

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🔷 Section D: Show me...

Create examples:

1) A sequence with a common ratio of 6
→ Pick any first term, say 1:
→ 1, 6, 36, 216, ...
✔ Example: 1, 6, 36, 216, ...

2) A decreasing geometric sequence
→ Ratio between 0 and 1, or negative with magnitude >1 but alternating?
Better: ratio < 1, e.g., 1/2
→ 16, 8, 4, 2, 1, ...
✔ Example: 16, 8, 4, 2, 1, ...

3) A sequence with a common ratio of -2
→ Start with 1: 1, -2, 4, -8, 16, ...
✔ Example: 1, -2, 4, -8, 16, ...

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✔ Final Summary:

#### Section A: Geometric Sequences (Circle):
- 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, ...

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

#### Section C: Fill 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. 16, 8, 4, 2, 1, ...
3. 1, -2, 4, -8, 16, ...

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✔ All problems solved! Let me know if you'd like this formatted as a printable answer sheet.