Math Problem Statement
Determine if the following sequence is arithmetic, geometric, or neither. Write JUST for arithmetic, DO for geometric, and IT for neither.
- a,ar,ar^2,ar^3, ...
- -1, 1, -1, 1, ...
- 1,1,2,3,5,8 ...
- 40,20,10,5, ...
- 1/2, 2/3 , 3/4, 4/5 , ...
- x, x+y , x+2y , x+3y, ...
- -3, -3/2 , 0, ...
- 1,3,5,7 ...
- 2,6,18,54,162, ...
- 3,6,12,24, ...
Solution
Here are the sequences classified as requested:
- DO (This is a geometric sequence: )
- IT (This sequence is neither arithmetic nor geometric: )
- IT (This is the Fibonacci sequence, which is neither arithmetic nor geometric: )
- DO (This is a geometric sequence with a common ratio of : )
- IT (This sequence is neither arithmetic nor geometric: )
- JUST (This is an arithmetic sequence with a common difference of : )
- JUST (This is an arithmetic sequence with a common difference of : )
- JUST (This is an arithmetic sequence with a common difference of 2: )
- DO (This is a geometric sequence with a common ratio of 3: )
- DO (This is a geometric sequence with a common ratio of 2: )
Would you like more details on any of these classifications or have any questions?
Here are five related questions to deepen your understanding:
- What defines an arithmetic sequence, and how do you find its common difference?
- How do you identify a geometric sequence and determine its common ratio?
- What are some examples of sequences that are neither arithmetic nor geometric?
- How do the Fibonacci sequence and other special sequences differ from standard arithmetic or geometric sequences?
- 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.
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Geometric Sequences
Fibonacci Sequence
Series and Patterns
Formulas
Arithmetic sequence formula: a_n = a_1 + (n-1)d
Geometric sequence formula: a_n = a_1 * r^(n-1)
Theorems
Properties of arithmetic sequences
Properties of geometric sequences
Characteristics of the Fibonacci sequence
Suitable Grade Level
Grades 9-12