Exponential Functions/Geometric Sequences Review

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent.

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What is a geometric sequence?

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A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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What is the common ratio in a geometric sequence?

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The common ratio is the factor by which we multiply each term to get the next term in a geometric sequence.

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How do you find the nth term of a geometric sequence?

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The nth term of a geometric sequence can be found using the formula a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

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What is the formula for the sum of the first n terms of a geometric series?

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The sum S_n of the first n terms of a geometric series is given by S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term and r is the common ratio.

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What is an asymptote?

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An asymptote is a line that a graph approaches but never touches or crosses. In exponential functions, horizontal asymptotes indicate the value that the function approaches as x approaches infinity.

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What does it mean for a function to exhibit exponential growth?

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Exponential growth occurs when a quantity increases by a consistent percentage over equal time intervals, resulting in a rapid increase.

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What does it mean for a function to exhibit exponential decay?

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Exponential decay occurs when a quantity decreases by a consistent percentage over equal time intervals, resulting in a rapid decrease.

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How do you determine if a sequence is exponential growth or decay?

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If the common ratio is greater than 1, the sequence exhibits exponential growth. If the common ratio is between 0 and 1, it exhibits exponential decay.

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What is the explicit rule for the sequence 5, 15, 45, ...?

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The explicit rule is a_n = 5(3)^(n-1).