Geometric Sequences Worksheet | PDF Printable Algebra Worksheet - Free Printable

Let’s go step by step through each section of the worksheet. I’ll solve everything carefully and check my work before giving the final answer.

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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 check each:

1. 1, 1, 2, 3, 5, 8, ... → This is Fibonacci — adds previous two terms → ✘ Not geometric
2. 6000, 3000, 1500, ... → Divide by 2 each time → ✔ Geometric (ratio = 0.5)
3. 1, 3, 6, 10, 15, ... → Adds increasing numbers → triangular numbers → ✘ Not geometric
4. 1, 1/3, 1/4, 1/8, ... → From 1 to 1/3 is ×1/3; 1/3 to 1/4 is ×3/4? No consistent ratio → ✘
Wait — let me recalculate:
1 → 1/3 = ×1/3
1/3 → 1/4 = ×(1/4)/(1/3) = 3/4 → not same → ✘
5. -8, -16, -32, -64, ... → Multiply by 2 each time → ✔ Geometric (ratio = 2)
6. x, x+1, x+2, x+3, ... → Arithmetic → ✘
7. 10, 100, 1000, 10000, ... → Multiply by 10 → ✔ Geometric
8. -1, 1, -1, 1, -1, ... → Multiply by -1 → ✔ Geometric
9. 4, 6, 9, 13.5, ... → 6/4=1.5, 9/6=1.5, 13.5/9=1.5 → ✔ Geometric (ratio=1.5)
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 → ✔ Geometric

✔ 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 _______________

Actually — note: the worksheet says “geometric series” but in context it means “geometric sequence”. In math, a *sequence* is just the list, a *series* is the sum. But since this is for school level, they probably mean sequence.

So:
A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio.

But since the blank is short, maybe:
→ “has a constant ratio between consecutive terms.”

I’ll write that.

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Section B: Find the common ratio

Common ratio = second term ÷ first term (and should be same for all adjacent pairs)

1) 5, 20, 80, 320 → 20÷5 = 4 → ✔ r=4
2) 1, -5, 25, -125 → -5÷1 = -5 → ✔ r=-5
3) 3, 4.5, 6.75, 10.125 → 4.5÷3 = 1.5 → ✔ r=1.5
4) 3.2, 6.4, 12.8, 25.6 → 6.4÷3.2 = 2 → ✔ r=2
5) 6000, 600, 60, 6 → 600÷6000 = 0.1 → ✔ r=0.1
6) 1, ?, 9, ?, 81 → Let’s find ratio from 1 to 9 over 2 steps: so r² = 9 → r=3 or r=-3. Since 81 is positive, likely r=3. Check: 1×3=3, 3×3=9, 9×3=27, 27×3=81 → yes → r=3
7) 1, 1/3, 1/9, 1/27 → (1/3)/1 = 1/3 → ✔ r=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 — 0.125 ÷ 0.4 = 125/400 = 5/16 = 0.3125 → not 0.2! Mistake?

Wait — let's calculate again:

10 → 2 = ×0.2
2 → 0.4 = ×0.2
0.4 → ? If ×0.2 = 0.08, but given is 0.125 → that doesn’t match.

Wait — maybe typo? Or did I misread?

Given: 10, 2, 0.4, 0.125

Check ratios:

2 / 10 = 0.2
0.4 / 2 = 0.2
0.125 / 0.4 = 0.3125 → not equal → contradiction?

Wait — perhaps it’s 0.08? But written as 0.125? Hmm.

Alternatively, maybe it’s 10, 2, 0.4, 0.08 — then r=0.2.

But the problem says 0.125. Let me double-check calculation:

0.4 × 0.2 = 0.08, not 0.125.

Unless... maybe it’s not decimal? 0.125 is 1/8.

Wait — let’s try fractions:

10 = 10/1
2 = 2/1
0.4 = 2/5
0.125 = 1/8

Ratios:

2/10 = 1/5
(2/5)/2 = (2/5)*(1/2)=1/5
(1/8)/(2/5) = (1/8)*(5/2)=5/16 ≠ 1/5 → still not matching.

This suggests either an error in the problem or I’m missing something.

Wait — look back at original image description — user said “uploaded an image”, but I don’t have access to image — only text provided in prompt.

In the initial problem statement, under Section B #8: “10, 2, 0.4, 0.125, …”

Perhaps it’s a trick? Or maybe it’s 0.08? Because 0.4 × 0.2 = 0.08.

Maybe it’s a typo in the problem? For educational purposes, if we assume it’s meant to be geometric, then likely 0.08 was intended.

But let’s see: 10, 2, 0.4, 0.08, 0.016 — that would be r=0.2.

Alternatively, maybe 0.125 is correct and ratio changes? But then not geometric.

Another idea: perhaps it’s 10, 2, 0.4, 0.08 — and 0.125 is a mistake.

Since this is a standard worksheet, and other problems are clean, I suspect it’s supposed to be 0.08.

But wait — let me check online or think differently.

Actually, 0.125 is 1/8, and 0.4 is 2/5.

From 2 to 0.4 is ÷5, from 0.4 to 0.125 is ×5/16? Not consistent.

Perhaps the sequence is 10, 2, 0.4, 0.08 — and "0.125" is a typo for "0.08".

I’ll proceed with r=0.2 assuming typo, because otherwise it’s not geometric.

But let’s hold on — maybe I miscalculated 0.125 / 0.4.

0.125 ÷ 0.4 = 125/400 = 25/80 = 5/16 = 0.3125 — yes.

Not 0.2.

Alternative approach: maybe the ratio is not constant? But the instruction is to find common ratio for geometric sequences — implying they are geometric.

Perhaps for #8, it’s not geometric? But the section says “find the common ratio of the geometric sequences” — so all listed are assumed geometric.

This is confusing.

Wait — let’s calculate backwards: if last term is 0.125, and previous is 0.4, ratio is 0.125/0.4=0.3125.

Then before that: 0.4 / 0.3125 = 1.28, but we have 2 — not matching.

If ratio is r, then:

Term1: 10
Term2: 10r = 2 → r=0.2
Term3: 10r^2 = 10*(0.04)=0.4 → good
Term4: 10r^3 = 10*(0.008)=0.08 — but given 0.125 — conflict.

So unless the fourth term is 0.08, it’s not geometric.

Given that this is a school worksheet, and likely has no errors, I think there might be a typo in the problem as presented to me. Perhaps in the actual image, it’s 0.08.

To resolve, I’ll assume it’s 0.08 and r=0.2. Otherwise, it’s not geometric, which contradicts the section title.

For now, I'll put r=0.2 for #8, noting the discrepancy.

9) x, x², x³, x⁴ → ratio = x²/x = x → ✔ r=x
10) -7, -14, -28, -56, -112 → -14/-7=2, -28/-14=2, etc → ✔ r=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

We need to find missing terms using common ratio.

1) 2, __, 200, __, 20000

First, find ratio. From 2 to 200 is two steps: so 2 * r^2 = 200 → r^2 = 100 → r=10 or r=-10. Since all terms positive, r=10.

So:
2, 2*10=20, 200, 200*10=2000, 20000 → gaps: 20 and 2000

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

From 15 to 75 is ×5, so r=5.

Previous term: 15 / 5 = 3
Next term: 75 * 5 = 375
So: 3, 15, 75, 375

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

Ratio: 4/1=4 → r=4
Next: 4*4=16, 16*4=64 → so 16, 64

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

Three steps from 7 to 189: 7 * r^3 = 189 → r^3 = 27 → r=3

So:
7, 7*3=21, 21*3=63, 63*3=189 → gaps: 21, 63

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

From 200 to 50 is two steps: 200 * r^2 = 50 → r^2 = 50/200 = 1/4 → r=1/2 or r=-1/2. Terms positive, so r=0.5

So:
200, 200*0.5=100, 50, 50*0.5=25 → gaps: 100, 25

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

From 12 to -36 is ×(-3), so r=-3

Previous term: 12 / (-3) = -4
Next term: -36 * (-3) = 108
So: -4, 12, -36, 108

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

From first 8 to third term 8: two steps, so 8 * r^2 = 8 → r^2=1 → r=1 or r=-1

If r=1: 8,8,8,8 — possible
If r=-1: 8,-8,8,-8 — also possible

But the sequence is 8, __, 8, __ — so if r=-1: 8, -8, 8, -8 — fits
If r=1: 8,8,8,8 — also fits

Which one? Probably r=-1 because if r=1, it's trivial, and often such problems use alternating signs.

Check: from 8 to next: if r=-1, then -8, then 8, then -8 — yes.

So gaps: -8, -8

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

From 1/3 to 1/12 is two steps: (1/3) * r^2 = 1/12 → r^2 = (1/12)/(1/3) = (1/12)*3 = 1/4 → r=1/2 or r=-1/2

Terms positive, so r=1/2

So:
1/3, (1/3)*(1/2)=1/6, 1/12, (1/12)*(1/2)=1/24 → gaps: 1/6, 1/24

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

Find ratio: 512 / 4096 = ? 4096 ÷ 8 = 512, so 512/4096 = 1/8 → r=1/8? But let's see:

4096 to 512 is ÷8, so r=1/8

Then next: 512 * (1/8) = 64
Then 64 * (1/8) = 8 — matches
Then 8 * (1/8) = 1

So sequence: 4096, 512, 64, 8, 1 → gaps: 64, 1

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

Ratio: -100 / -20 = 5 → r=5

Next: -100 * 5 = -500
Then -500 * 5 = -2500
So gaps: -500, -2500

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Section D: Show me...

1) A sequence with common ratio of 6
Start with any number, say 1: 1, 6, 36, 216, ...
Or 2: 2, 12, 72, 432, ...
Any is fine. I'll use 1,6,36,...

2) A decreasing geometric sequence
Decreasing means |r| < 1 and positive terms, or negative ratio with absolute value >1 but alternating — but usually "decreasing" implies getting smaller in magnitude if positive.

Example: 100, 50, 25, 12.5, ... (r=0.5)
Or 8, 4, 2, 1, ... (r=0.5)

3) A sequence with common ratio of -2
Start with 1: 1, -2, 4, -8, 16, ...
Or 3: 3, -6, 12, -24, ...

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Now, compiling all answers.

Final Answer must include all parts.

Let me organize:

Section A:
Circled 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 sequence has a constant ratio between consecutive terms.

Section B:
1) 4
2) -5
3) 1.5
4) 2
5) 0.1
6) 3
7) \frac{1}{3}
8) 0.2 [note: assuming typo, should be 0.08 instead of 0.125]
9) x
10) 2

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

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

Now, for Final Answer, since it's multiple parts, I need to present clearly.

The instruction says: "Provide only the final correct answer — this should come AFTER all explanations and reasoning."

And "Complete ALL reasoning and explanations BEFORE writing the Final Answer."

So I've done reasoning above. Now Final Answer should summarize all answers concisely.

Since it's a worksheet, perhaps list by section.

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 completion: "has a constant ratio between consecutive terms."

Section B:
1) 4
2) -5
3) 1.5
4) 2
5) 0.1
6) 3
7) \frac{1}{3}
8) 0.2
9) x
10) 2

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

Section D:
1) e.g., 1, 6, 36, 216, ...
2) e.g., 100, 50, 25, 12.5, ...
3) e.g., 1, -2, 4, -8, 16, ...

Note: For Section B #8, if the term is indeed 0.125, then it's not geometric, but based on context, 0.2 is expected.

I think this covers everything.