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Questions and Answers
What is the formula for the nth term in an arithmetic progression?
What is the common difference in the arithmetic progression 2, 5, 8, 11, 14?
What is the purpose of the formula for the nth term in an arithmetic progression?
What is an example of a real-world problem that can be modeled using an arithmetic progression?
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What is the term for the sum of a finite number of terms in an arithmetic progression?
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What is the characteristic of an arithmetic progression that makes it useful for modeling real-world problems?
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What is the definition of an arithmetic mean?
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What is the term for a sequence that extends indefinitely in one direction?
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Study Notes
Definition
- An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term.
- The fixed constant is called the common difference (d).
Formula
- The general formula for an arithmetic progression is: an = a1 + (n - 1)d where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Key Properties
- Common difference (d) is the same throughout the sequence.
- The sequence progresses by adding the common difference to the previous term.
- The nth term can be calculated using the formula an = a1 + (n - 1)d.
Examples
- 2, 5, 8, 11, 14... (a1 = 2, d = 3)
- 10, 15, 20, 25, 30... (a1 = 10, d = 5)
Applications
- Calculating the sum of an arithmetic series.
- Modeling real-world problems, such as:
- Distance traveled by an object moving at a constant acceleration.
- Increase in population size over time.
- Compound interest on an investment.
Important Concepts
- Arithmetic mean: The average of two or more numbers in an arithmetic progression.
- Arithmetic series: The sum of a finite number of terms in an arithmetic progression.
- Infinite arithmetic progression: A sequence that extends indefinitely in one direction.
Definition
- An arithmetic progression (AP) is a sequence of numbers where each term is obtained by adding a fixed constant, called the common difference (d), to the previous term.
Formula
- The general formula for an arithmetic progression is: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Key Properties
- The common difference (d) remains the same throughout the sequence, allowing the sequence to progress by adding the common difference to the previous term.
- The nth term can be calculated using the formula an = a1 + (n - 1)d.
Examples
- The sequence 2, 5, 8, 11, 14... has a first term (a1) of 2 and a common difference (d) of 3.
- The sequence 10, 15, 20, 25, 30... has a first term (a1) of 10 and a common difference (d) of 5.
Applications
- Arithmetic progressions can be used to calculate the sum of an arithmetic series.
- APs can model real-world problems, such as:
- Distance traveled by an object moving at a constant acceleration.
- Increase in population size over time.
- Compound interest on an investment.
Important Concepts
- Arithmetic mean: The average of two or more numbers in an arithmetic progression.
- Arithmetic series: The sum of a finite number of terms in an arithmetic progression.
- Infinite arithmetic progression: A sequence that extends indefinitely in one direction.
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Description
Learn about arithmetic progression, its formula, and key properties. Understand how to find the nth term using the common difference and first term.
