Comprehensive worksheet for practicing geometric sequences, featuring sections on identifying sequences, calculating common ratios, filling gaps, and creating sequences with given ratios.

Let’s solve this Geometric Sequences worksheet step by step.

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## ✔ Section A: Circle all the geometric sequences below.

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

We’ll check each sequence:

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Row 1:

- 1, 1, 2, 3, 5, 8, ... → Fibonacci (additive), not geometric
- 6000, 3000, 1500, ... → ÷2 each time → geometric (r = 0.5)
- 1, 3, 6, 10, 15, ... → Triangular numbers (add increasing integers), not geometric

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Row 2:

- 1, 1/3, 1/4, 1/8, ... → 1→1/3 (×1/3), 1/3→1/4 (×3/4) → not constant ratio → not geometric
- -8, -16, -32, -64, ... → ×2 each time → geometric (r = 2)
- x, x+1, x+2, x+3, ... → Adding 1 → arithmetic, not geometric

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Row 3:

- 10, 100, 1000, 10000, ... → ×10 → geometric (r = 10)
- -1, 1, -1, 1, -1, ... → ×(-1) → geometric (r = -1)
- 4, 6, 9, 13.5, ... → 4×1.5=6, 6×1.5=9, 9×1.5=13.5 → geometric (r = 1.5)

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Row 4:

- 5, 10, 15, 20, ... → +5 → arithmetic, not geometric
- 0.1, 0.2, 0.3, 0.4, ... → +0.1 → arithmetic, not geometric
- a, 2a, 4a, 8a, ... → ×2 → geometric (r = 2)

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✔ Circled 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, ...

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Now finish the sentence:

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

*(Note: The question says “geometric series”, which technically is the sum, but sometimes used interchangeably with “sequence” in basic contexts. But strictly speaking:)*

✔ Answer:
A geometric series is the sum of the terms of a geometric sequence.

*(If they meant “sequence”, then: "is a sequence where each term is obtained by multiplying the previous term by a constant common ratio.")*

But since it says “series”, we go with the sum definition.

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

Common ratio `r = term₂ / term₁`

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1) 5, 20, 80, 320, ...
r = 20/5 = 4

2) 1, -5, 25, -125, 625, ...
r = -5/1 = -5

3) 3, 4.5, 6.75, 10.125, ...
r = 4.5/3 = 1.5

4) 3.2, 6.4, 12.8, 25.6, ...
r = 6.4/3.2 = 2

5) 6000, 600, 60, 6, ...
r = 600/6000 = 0.1

6) 1, ?, 9, ?, 81, ...
Let’s assume positions: a₁=1, a₃=9 → r² = 9/1 → r = 3 or -3
Since 81 is positive and 1→9→81, likely r = 3

7) 1, 1/3, 1/9, 1/27, ...
r = (1/3)/1 = 1/3

8) 10, 2, 0.4, 0.125, ...
Wait — 10→2 = ×0.2, 2→0.4 = ×0.2, but 0.4→0.125 = ×0.3125 → Not geometric?
❗️Wait — 0.4 × 0.2 = 0.08, not 0.125 → So this is NOT geometric?
But the worksheet says “Find the common ratio of the geometric sequences.” So maybe typo?

Let’s double-check:
10 → 2 → 0.4 → next should be 0.08, but it says 0.125 → so this is NOT geometric.
Perhaps misprint? Or maybe 0.125 is wrong?
If we assume it’s supposed to be 0.08, then r = 0.2.
But as written, it’s not geometric. However, since the worksheet asks for it, perhaps we assume it's intended to be geometric and correct it?

Alternatively, maybe 0.125 is a mistake — let’s proceed assuming it’s meant to be 0.08 → r = 0.2

*(In many worksheets, this is a common error — actual sequence should be 10, 2, 0.4, 0.08, 0.016...)*

So we’ll say r = 0.2

9) x, x², x³, x⁴, ...
r = x²/x = x

10) -7, -14, -28, -56, -112, ...
r = (-14)/(-7) = 2

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✔ 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 in last term)*
9) x
10) 2

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## ✔ Section C: Fill the gaps in these geometric sequences.

Use: next term = previous × r

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1) 2, __, 200, __, 20000, ...

Let’s find r: from 2 to 200 → 2 × r² = 200 → r² = 100 → r = 10 or -10
Assume positive: r = 10
So: 2, 20, 200, 2000, 20000

✔ Answer: 20, 2000

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2) __, 15, 75, __, ...

From 15 to 75 → r = 75/15 = 5
So previous term: 15 / 5 = 3
Next term: 75 × 5 = 375

✔ Answer: 3, 375

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3) 1, 4, __, __, ...

r = 4/1 = 4
So: 1, 4, 16, 64

✔ Answer: 16, 64

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4) 7, __, __, 189, ...

Let’s say 7 × r³ = 189 → r³ = 189/7 = 27 → r = 3
So: 7, 21, 63, 189

✔ Answer: 21, 63

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5) 200, __, 50, __, ...

200 → 50 → divided by 4 → so r² = 50/200 = 1/4 → r = 1/2 or -1/2
Assume positive: r = 0.5
So: 200, 100, 50, 25

✔ Answer: 100, 25

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6) __, 12, -36, __, ...

From 12 to -36 → r = -36/12 = -3
Previous term: 12 / (-3) = -4
Next term: -36 × (-3) = 108

✔ Answer: -4, 108

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7) 8, __, 8, __, ...

First term 8, third term 8 → 8 × r² = 8 → r² = 1 → r = 1 or -1
If r = 1 → sequence: 8, 8, 8, 8 — possible
If r = -1 → 8, -8, 8, -8 — also possible
But since it says “fill the gaps”, and both are valid, but often r=-1 is more interesting.

Check: if r = -1 → 8, -8, 8, -8

✔ Answer: -8, -8 (if r=-1)
Or 8, 8 (if r=1)

But since second term is blank and third is 8, likely alternating → r = -1

✔ Final: -8, -8

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8) 1/3, __, 1/12, __, ...

Let’s find r: (1/12) / (1/3) = (1/12) × 3 = 1/4 → so r² = 1/4 → r = 1/2 or -1/2
Assume positive: r = 1/2
Then: 1/3, (1/3)(1/2)=1/6, 1/12, (1/12)(1/2)=1/24

✔ Answer: 1/6, 1/24

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9) 4096, 512, __, 8, __, ...

From 4096 → 512 → r = 512/4096 = 1/8
Check: 512 × 1/8 = 64, then 64 × 1/8 = 8, then 8 × 1/8 = 1

✔ Answer: 64, 1

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10) -20, -100, __, __, ...

r = (-100)/(-20) = 5
So: -20, -100, -500, -2500

✔ Answer: -500, -2500

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✔ Section C Answers:

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

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## ✔ Section D: Show me...

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1) A sequence with a common ratio of 6

Start with any number, say 1:
1, 6, 36, 216, 1296, ...

✔ Answer: 1, 6, 36, 216, ...

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2) A decreasing geometric sequence

Means |r| < 1 and r > 0, or r negative but magnitude <1? Actually, “decreasing” usually means each term smaller than previous (numerically).

Examples:

- 16, 8, 4, 2, 1, ... (r = 0.5)
- 100, 50, 25, 12.5, ... (r = 0.5)

✔ Answer: 16, 8, 4, 2, 1, ...

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3) A sequence with a common ratio of -2

Start with 1:
1, -2, 4, -8, 16, ...

✔ Answer: 1, -2, 4, -8, 16, ...

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## 🎯 FINAL SUMMARY:

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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 of a geometric sequence.*

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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

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Section C: Fill the 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

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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 done! You’ve mastered geometric sequences! 🎉