Determine if the following sequence is arithmetic, geometric, or neither.
Write
JUST for arithmetic, 
DO for geometric, and
IT for neither.

1. a,ar,ar^2,ar^3, ...
2. -1, 1, -1, 1, ...
3. 1,1,2,3,5,8 ...
4. 40,20,10,5, ...
5. 1/2, 2/3 , 3/4, 4/5 , ...
6. x, x+y , x+2y , x+3y, ...
7. -3, -3/2 , 0, ...
8. 1,3,5,7 ...
9. 2,6,18,54,162, ...
10. 3,6,12,24, ...

Here are the sequences classified as requested:

1. **DO** (This is a geometric sequence: \( a, ar, ar^2, ar^3, \dots \))
2. **IT** (This sequence is neither arithmetic nor geometric: \( -1, 1, -1, 1, \dots \))
3. **IT** (This is the Fibonacci sequence, which is neither arithmetic nor geometric: \( 1, 1, 2, 3, 5, 8, \dots \))
4. **DO** (This is a geometric sequence with a common ratio of \( \frac{1}{2} \): \( 40, 20, 10, 5, \dots \))
5. **IT** (This sequence is neither arithmetic nor geometric: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots \))
6. **JUST** (This is an arithmetic sequence with a common difference of \( y \): \( x, x+y, x+2y, x+3y, \dots \))
7. **JUST** (This is an arithmetic sequence with a common difference of \( \frac{3}{2} \): \( -3, -\frac{3}{2}, 0, \dots \))
8. **JUST** (This is an arithmetic sequence with a common difference of 2: \( 1, 3, 5, 7, \dots \))
9. **DO** (This is a geometric sequence with a common ratio of 3: \( 2, 6, 18, 54, 162, \dots \))
10. **DO** (This is a geometric sequence with a common ratio of 2: \( 3, 6, 12, 24, \dots \))

Would you like more details on any of these classifications or have any questions?

Here are five related questions to deepen your understanding:
1. What defines an arithmetic sequence, and how do you find its common difference?
2. How do you identify a geometric sequence and determine its common ratio?
3. What are some examples of sequences that are neither arithmetic nor geometric?
4. How do the Fibonacci sequence and other special sequences differ from standard arithmetic or geometric sequences?
5. Can a sequence be both arithmetic and geometric? If so, under what conditions?

**Tip:** Always check the ratio or difference between consecutive terms to determine the type of sequence.

Determine if the following sequence is arithmetic, geometric, or neither. Write JUST for arithmetic, DO for geometric, and IT for neither. 1. a, ar, ar^2, ar^3, ... 2. -1, 1, -1, 1, ... 3. 1,1,2,3,5,8 ... 4. 40,20,10,5, ... 5. 1/2, 2/3 , 3/4, 4/5 , ... 6. x, x+y , x+2y , x+3y, ... 7. -3, -3/2 , 0, ... 8. 1,3,5,7 ... 9. 2,6,18,54,162, ... 10. 3,6,12,24, ...

Classify various sequences as arithmetic, geometric, or neither. This problem involves understanding the arithmetic sequence formula, geometric sequence formula, and recognizing patterns like the Fibonacci sequence. Suitable for students in Grades 9-12.

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