Are the following sequences arithmetic or geometric? a) 4, 8, 16, 32, ... b) 35, 27, 19, 11, ... c) 5, 8, 11, 14, ...

Understand the Problem

The question is asking to determine whether each of the provided sequences is arithmetic or geometric based on the definitions provided, focusing on the operations of addition/subtraction and multiplication/division.

Answer

a) Geometric b) Arithmetic c) Arithmetic

Steps to Solve

Identify the first sequence: 4, 8, 16, 32, ...

Examine the differences and ratios between successive terms.
- Difference between terms:
  - $8 - 4 = 4$
  - $16 - 8 = 8$
  - $32 - 16 = 16$
The differences are not constant, so it's not arithmetic.
- Ratios of terms:
  - $\frac{8}{4} = 2$
  - $\frac{16}{8} = 2$
  - $\frac{32}{16} = 2$
The ratios are constant, indicating it's a geometric sequence with a common ratio of 2.
Identify the second sequence: 35, 27, 19, 11, ...

Again, analyze the differences:
- Difference between terms:
  - $27 - 35 = -8$
  - $19 - 27 = -8$
  - $11 - 19 = -8$
The differences are constant (-8), confirming it is an arithmetic sequence.
Identify the third sequence: 5, 8, 11, 14, ...

Check the differences to determine its type:
- Difference between terms:
  - $8 - 5 = 3$
  - $11 - 8 = 3$
  - $14 - 11 = 3$
The differences are constant (3), indicating it is also an arithmetic sequence.

a) Geometric sequence
b) Arithmetic sequence
c) Arithmetic sequence

More Information

The first sequence doubles each term, fitting the definition of a geometric sequence.
The second and third sequences subtract a constant value from each term, fitting the definition of an arithmetic sequence.

Tips

A common mistake is to confuse the difference between terms as being constant in a geometric sequence; they should be ratios instead.
The opposite can occur as well; one might look for constant ratios in an arithmetic sequence rather than checking the differences.

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