Geometric sequences, exponential growth and decay, logarithm properties, modeling and function composition and inverses.
40 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a geometric sequence?
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A sequence where each term is found by multiplying the previous term by a constant ratio- How do you find the common ratio in a geometric sequence?
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Divide any term by the previous term- What is the formula for the nth term of a geometric sequence?
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a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio- What is the formula for the sum of a finite geometric series?
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S_n = a_1(1 - r^n) / (1 - r), when r is not equal to 1- When does an infinite geometric series converge?
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When the common ratio r satisfies |r| < 1- What is the sum formula for an infinite geometric series when |r| < 1?
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S = a_1 / (1 - r)- In a geometric sequence with r > 1, how do the terms behave?
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They increase exponentially in magnitude- In a geometric sequence with 0 < r < 1, how do the terms behave?
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They decrease and approach zero- What is an exponential function?
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A function of the form f(x) = a * b^x where a is nonzero, b > 0, and b ≠ 1- What does the base b determine in f(x) = a * b^x?
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Whether the function grows (b > 1) or decays (0 < b < 1), and the rate of change- What is the natural exponential function?
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f(x) = e^x, where e is approximately 2.71828- When does f(x) = a * b^x represent exponential growth?
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When a > 0 and b > 1- When does f(x) = a * b^x represent exponential decay?
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When a > 0 and 0 < b < 1- What is half-life?
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The time it takes for a quantity to decrease to half its original amount- What is doubling time?
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The time it takes for a quantity to increase to twice its original amount
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