Geometric Sequences Worksheet | PDF Printable Algebra Worksheet - Free Printable

Let’s go step by step through each section.

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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 every time to get the next term. That number is called the “common ratio”.

Let’s check each:

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, ... → Adding increasing numbers (triangular numbers). Not geometric. ✘
4. 1, 1/3, 1/4, 1/8, ... → From 1 to 1/3 is ×1/3, but 1/3 to 1/4 is not same ratio. ✘
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, ... → Add 1 each time → arithmetic, not geometric. ✘
7. 10, 100, 1000, 10000, ... → Multiply by 10 each time → YES ✔
8. -1, 1, -1, 1, -1, ... → Multiply by -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, ... → Multiply by 2 each time → YES ✔

✔ So circle these:
- 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. But since this worksheet says “series” and we’ve been talking about sequences, it might be a typo — they probably mean “sequence”. In context, likely:
→ *A geometric sequence has a constant ratio between consecutive terms.*

But let’s stick with what’s asked: “A geometric series ___”
Actually, in math, a “geometric series” is the SUM of a geometric sequence. Example: 1 + 2 + 4 + 8 + ...
So:
A geometric series is the sum of the terms of a geometric sequence.

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Section B: Find the common ratio

Common ratio = second term ÷ first term (and should be same for any two consecutive terms)

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 or 3/2
4) 3.2, 6.4, 12.8 → 6.4÷3.2 = 2
5) 6000, 600, 60 → 600÷6000 = 0.1 or 1/10
6) 1, ?, 9, ?, 81 → Since 1 to 9 is two steps, so ratio squared = 9 → ratio = √9 = 3 (since 1×3=3, 3×3=9, etc.)
7) 1, 1/3, 1/9, 1/27 → (1/3)÷1 = 1/3
8) 10, 2, 0.4, 0.125 → 2÷10 = 0.2 or 1/5
9) x, x², x³, x⁴ → x² ÷ x = x
10) -7, -14, -28 → -14 ÷ -7 = 2

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Section C: Fill the gaps

We need to find missing terms using the common ratio.

1) 2, __, 200, __, 20000
From 2 to 200: that’s two steps → 2 × r² = 200 → r² = 100 → r = 10
So: 2, 20, 200, 2000, 20000

2) __, 15, 75, __
15 to 75 → ×5 → so before 15: 15÷5 = 3; after 75: 75×5 = 375

3) 1, 4, __, __
Ratio = 4÷1 = 4 → next: 4×4=16, then 16×4=64

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

5) 200, __, 50, __
200 to 50 is two steps → 200 × r² = 50 → r² = 50/200 = 1/4 → r = 1/2 or -1/2
Since 200 to 50 is decreasing, likely positive ratio 1/2? Wait: 200 × (1/2) = 100, 100 × (1/2) = 50 → yes. Then next: 50 × 1/2 = 25
But could also be negative? If r = -1/2: 200 × (-1/2) = -100, -100 × (-1/2) = 50 → also works!
Problem doesn’t specify, but usually assume positive unless signs alternate. Here no sign change → use r = 1/2
So: 200, 100, 50, 25

Wait — looking at problem 6 later has alternating signs, so here probably positive. Let’s go with 100 and 25.

6) __, 12, -36, __
12 to -36 → ×(-3) → so before 12: 12 ÷ (-3) = -4; after -36: -36 × (-3) = 108

7) 8, __, 8, __
If middle term is same as first, maybe ratio is 1? But then all terms 8. Or ratio -1? 8, -8, 8, -8 → fits!
Check: 8 × (-1) = -8, -8 × (-1) = 8 → yes. So: 8, -8, 8, -8

8) 1/3, __, 1/12, __
From 1/3 to 1/12 is two steps → (1/3) × r² = 1/12 → r² = (1/12)/(1/3) = 1/4 → r = 1/2 or -1/2
Try r = 1/2: 1/3 × 1/2 = 1/6, 1/6 × 1/2 = 1/12 → good. Next: 1/12 × 1/2 = 1/24
Or r = -1/2: 1/3 × (-1/2) = -1/6, -1/6 × (-1/2) = 1/12 → also works.
Again, no sign change shown, so probably positive. Use r=1/2.
So: 1/3, 1/6, 1/12, 1/24

9) 4096, 512, __, 8, __
4096 to 512: 512 ÷ 4096 = 1/8 → ratio = 1/8? Check: 512 × (1/8) = 64, 64 × (1/8) = 8 → yes. Then 8 × (1/8) = 1
So: 4096, 512, 64, 8, 1

10) -20, -100, __, __
-100 ÷ -20 = 5 → ratio = 5
Next: -100 × 5 = -500, then -500 × 5 = -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, ...
(You can start with any number, e.g., 2, 12, 72, ...)

2) A decreasing geometric sequence → Ratio between 0 and 1, or negative with absolute value >1? Decreasing means getting smaller.
Example: 100, 50, 25, 12.5, ... (ratio 0.5)
Or: 10, -20, 40, -80... but that alternates and magnitude increases — not decreasing.
Better: 81, 27, 9, 3, 1... (ratio 1/3) → decreasing.

3) Common ratio -2 → Start with 1: 1, -2, 4, -8, 16, ...

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Final Answer:

Section A – 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, ...

Finish the sentence:
A geometric series is the sum of the terms of a geometric sequence.

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

Section C – Fill gaps:
1) 2, 20, 200, 2000, 20000
2) 3, 15, 75, 375
3) 1, 4, 16, 64
4) 7, 21, 63, 189
5) 200, 100, 50, 25
6) -4, 12, -36, 108
7) 8, -8, 8, -8
8) 1/3, 1/6, 1/12, 1/24
9) 4096, 512, 64, 8, 1
10) -20, -100, -500, -2500

Section D – Show me:
1) Example: 1, 6, 36, 216, ...
2) Example: 100, 50, 25, 12.5, ...
3) Example: 1, -2, 4, -8, 16, ...