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Questions and Answers
What is a sequence?
What is a sequence?
A collection of numbers that follow a particular pattern.
What is an arithmetic sequence?
What is an arithmetic sequence?
A sequence with a common difference between two consecutive numbers constant.
What is a geometric sequence?
What is a geometric sequence?
A sequence with a common ratio between two consecutive numbers constant.
What is the common difference in an arithmetic sequence?
What is the common difference in an arithmetic sequence?
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What is the common ratio in a geometric sequence?
What is the common ratio in a geometric sequence?
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What is the formula for an arithmetic sequence?
What is the formula for an arithmetic sequence?
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How do you find the 4th term in the arithmetic sequence 3, 6, 9,...?
How do you find the 4th term in the arithmetic sequence 3, 6, 9,...?
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What is the formula for a geometric sequence?
What is the formula for a geometric sequence?
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How would you find the 10th term in the geometric sequence 2, 4, 8, 16,...?
How would you find the 10th term in the geometric sequence 2, 4, 8, 16,...?
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What is an arithmetic series?
What is an arithmetic series?
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What is the formula for an arithmetic series?
What is the formula for an arithmetic series?
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What is a geometric series?
What is a geometric series?
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When is a geometric series considered infinite?
When is a geometric series considered infinite?
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What does Sigma(Σ) represent?
What does Sigma(Σ) represent?
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How many terms are in the sequence 18, 22, 26, 30,..., 110?
How many terms are in the sequence 18, 22, 26, 30,..., 110?
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How many multiples of 5 are between 300 and 1000?
How many multiples of 5 are between 300 and 1000?
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What is the common difference of the sequence 13, 10.9, 8.8, 6.7,...?
What is the common difference of the sequence 13, 10.9, 8.8, 6.7,...?
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What is the common ratio of the sequence 3.1, 9.3, 27.9,...?
What is the common ratio of the sequence 3.1, 9.3, 27.9,...?
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How many rows and bricks are in the pile with 85 at the bottom and 1 at the top?
How many rows and bricks are in the pile with 85 at the bottom and 1 at the top?
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What is the sum of the first 6 terms in the geometric sequence 5, 10, 20,...?
What is the sum of the first 6 terms in the geometric sequence 5, 10, 20,...?
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What is the sum of the first 25 even integers?
What is the sum of the first 25 even integers?
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When will Josh's and Sophia's money sum up to 200?
When will Josh's and Sophia's money sum up to 200?
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What is the sum of the series 100 + 25 + 25/4 + 25/8 +...?
What is the sum of the series 100 + 25 + 25/4 + 25/8 +...?
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How far will a rubber ball travel before coming to rest if dropped from 729 cm?
How far will a rubber ball travel before coming to rest if dropped from 729 cm?
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Study Notes
Sequences
- A sequence is a collection of numbers arranged in a specific pattern, such as 2, 4, 8, 16, 32, which follows a doubling pattern.
Arithmetic Sequences
- Defined by a common difference between consecutive terms, e.g., 3, 6, 9, where the difference is consistently 3.
- The formula for finding the nth term is ( a_n = a_1 + d(n-1) ).
- An arithmetic series is the sum of terms and can be calculated using ( S_n = \frac{n}{2}(t_1 + t_n) ).
Geometric Sequences
- Characterized by a constant ratio between consecutive terms, like in 3, 9, 27, where the ratio is 3.
- The nth term can be calculated with the formula ( a_n = a_1(r)^{n-1} ).
- Geometric series are sums of terms, with finite series represented as ( S_n = t_1\frac{(1-r^n)}{(1-r)} ) and infinite series (|r| < 1) as ( S_{\infty} = \frac{t_1}{(1-r)} ).
Common Differences and Ratios
- Common difference refers to the value added in an arithmetic sequence, while common ratio is the multiplied value in a geometric sequence.
Recursive Formulas
- Recursive formulas for sequences involve using the previous term to determine the next term. For arithmetic: ( t_n = t_{n-1} + d ) and for geometric: ( t_n = t_{n-1}(r) ).
Sigma Notation
- Sigma (Σ) represents summation, allowing for easier expression of series.
Problem-Solving Techniques
- Use arithmetic sequence formulas to determine unknowns, such as the number of terms in a sequence. For example, finding terms between 18, 22, 26, and 110 involves reversing the arithmetic formula.
- Count multiples or terms by constructing arithmetic sequences and determining n terms using appropriate formulas.
Practice Examples
- Summing series can be practiced by applying the relevant formulas to determine the total sum of sequences or series, such as finding sums of the first 25 even integers or the total height traveled by a dropping ball.
Real-Life Applications
- Understanding sequences and series can help in financial planning, predicting trends in data, and calculating quantities in everyday scenarios.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge of sequences and series in Algebra 2 with these flashcards. Each card provides a definition and an example to enhance your understanding of important concepts like arithmetic and geometric sequences.
