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Questions and Answers
What is the common difference in the arithmetic sequence 6, 1, -4, -9, ...?
What is the common difference in the arithmetic sequence 6, 1, -4, -9, ...?
Which of the following sequences is classified as a geometric sequence?
Which of the following sequences is classified as a geometric sequence?
What is the formula for the nth term of an arithmetic sequence if the 0th term is 3 and the common difference is 4?
What is the formula for the nth term of an arithmetic sequence if the 0th term is 3 and the common difference is 4?
For the sequence defined by t_n = 4n - 2, what is the 5th term?
For the sequence defined by t_n = 4n - 2, what is the 5th term?
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Identify the common ratio of the geometric sequence 64, -32, 16, -8, 4, ...
Identify the common ratio of the geometric sequence 64, -32, 16, -8, 4, ...
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What type of graph do the points from an arithmetic sequence represent?
What type of graph do the points from an arithmetic sequence represent?
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If the nth term of a geometric sequence is given by t_n = t_0 ∙ r^n, what could r represent if r = -1/2?
If the nth term of a geometric sequence is given by t_n = t_0 ∙ r^n, what could r represent if r = -1/2?
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What would be the resulting sequence if the common difference is 2 and the first term is 7?
What would be the resulting sequence if the common difference is 2 and the first term is 7?
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Study Notes
Chapter 1 - Arithmetic and Geometric Sequences
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Objectives:
- Identify arithmetic and geometric sequences
- Find formulas for the nth term
What is a Sequence?
- A set of numbers, called terms, arranged in a specific order.
- Two basic types: arithmetic and geometric.
Arithmetic Sequences
- The difference between consecutive terms is constant.
- This constant difference is called the "common difference."
- Examples: 2, 6, 10, 14, 18... (common difference = 4)
- 17, 10, 3, -4, -11, -18... (common difference = -7)
- a, a+d, a+2d, a+3d, a+4d, ... (common difference = d)
Geometric Sequences
- The ratio of consecutive terms is constant.
- This constant ratio is called the "common ratio."
- Examples: 1, 3, 9, 27, 81... (common ratio = 3)
- 64, -32, 16, -8, 4... (common ratio = -1/2)
- a, ar, ar², ar³, ar⁴, ... (common ratio = r)
You Try!
- Identify the sequence type and the common difference or ratio.
- 5, 10, 20, 40... (geometric, ratio = 2)
- 6, 1, -4, -9... (arithmetic, difference = -5)
Notation
- 1st term: t₁; 2nd term: t₂; nth term: tₙ
- Some sequences are defined by rules or formulas.
- Example: tₙ = n² + 1
- t₁ = 1² + 1 = 2
- t₂ = 2² + 1 = 5
- Example: tₙ = n² + 1
Arithmetic Formulas
- nth term: tₙ = t₀ + dₙ
- t₀ is the 0th term (found by working backwards)
- d is the common difference, and n is the term number
Geometric Formulas
- nth term: tₙ = t₀ ⋅ rⁿ
- t₀ is the 0th term (found by working backwards)
- r is the common ratio, and n is the term number
Formal Definition
- A function whose domain is the set of positive integers.
- For example: tₙ = 4n - 2 (where n is a positive integer)
Graphing Sequences
- Write terms as ordered pairs (n, tₙ) and plot.
- Arithmetic sequences plot as points on a line
- Geometric sequences plot as points on an exponential curve.
Example 1
- Find the formula for the nth term of 3, 5, 7...
- Sketch the graph
Example 2
- Find the formula for the nth term of 3, 4.5, 6.75...
- Sketch the graph
You Try!
- Find the formula for the nth term of:
- 15, 7, -1, -9...
- 100, -50, 25, -12.5...
Example 3
- In a geometric sequence, t₃ = 12 and t₆ = 96. Find t₁₁.
Example 4
- In an arithmetic sequence, t₂ = 2 and t₅ = 16. Find t₁₀.
You Try!
- In a geometric sequence, t₂ = 2 and t₅ = 16. Find t₁₀.
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