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
Answer for screen readers
a) Geometric sequence
b) Arithmetic sequence
c) Arithmetic sequence
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.
- Difference between terms:
-
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.
- Difference between terms:
-
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.
- Difference between terms:
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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