Geometric Sequences worksheet with exercises on identifying, calculating ratios, and completing sequences.

Final Answer:
The geometric sequences are:
- $6000, 3000, 1500, \dots$
- $1, \frac{1}{3}, \frac{1}{4}, \frac{1}{8}, \dots$ → Not geometric (wait—check carefully!)
Actually, let’s verify each one properly:

A geometric sequence has a *constant ratio* between consecutive terms: each term = previous term × *r*.

Check each:

1. $1, 1, 2, 3, 5, 8, \dots$ → Fibonacci — not geometric ✘
2. $6000, 3000, 1500, \dots$ → $3000/6000 = 0.5$, $1500/3000 = 0.5$ → ✔1, 3, 6, 10, 15, \dots$ → triangular numbers — differences increase → ✘
4. $1, \frac{1}{3}, \frac{1}{4}, \frac{1}{8}, \dots$ → $\frac{1}{3}/1 = \frac{1}{3}$, $\frac{1}{4}/\frac{1}{3} = \frac{3}{4}$ ≠ $\frac{1}{3}$ → ✘
5. $-8, -16, -32, -64, \dots$ → $-16/-8 = 2$, $-32/-16 = 2$, etc. → ✔
6. $x, x+1, x+2, x+3, \dots$ → differences constant, not ratios → ✘
7. $10, 100, 1000, 10000, \dots$ → ×10 each time → ✔
8. $-1, 1, -1, 1, -1, \dots$ → ×(−1) each time → ✔
9. $4, 6, 9, 13.5, \dots$ → $6/4 = 1.5$, $9/6 = 1.5$, $13.5/9 = 1.5$ → ✔
10. $5, 10, 15, 20, \dots$ → +5 each time (arithmetic) → ✘
11. $0.1, 0.2, 0.3, 0.4, \dots$ → +0.1 → ✘
12. $a, 2a, 4a, 8a, \dots$ → ×2 each time → ✔

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

Now finish the sentence:
*A geometric series* is the sum of the terms of a geometric sequence.

But the question says “Circle all the geometric sequences below” — so final answer should list those sequences (as per standard expectation for this worksheet).

Since the instruction says “Provide only the final correct answer”, and it's a circle-all task, we list the ones that are geometric:

Final Answer:
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, …