Sequences and Exponential Functions Flashcard

Front

What is an arithmetic sequence?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference.

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How do you find the common difference in an arithmetic sequence?

Back

The common difference (d) can be found by subtracting any term from the subsequent term: d = a_{n+1} - a_n.

Front

What is the formula for the nth term of an arithmetic sequence?

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The nth term (a_n) of an arithmetic sequence can be calculated using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

Front

How do you calculate the sum of the first n terms of an arithmetic sequence?

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The sum (S_n) of the first n terms of an arithmetic sequence can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

Front

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.

Front

What is the base of an exponential function?

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The base of an exponential function is the constant (b) that is raised to the power of x. It determines the rate of growth or decay of the function.

Front

What transformation is applied to the function f(x) = 5^x to obtain g(x) = 5^{-x}?

Back

The transformation applied is a reflection over the y-axis.

Front

What is the significance of the point (0, 1) in the graph of an exponential function?

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The point (0, 1) represents the value of the function when x = 0, which is always 1 for any exponential function of the form f(x) = a * b^x.

Front

How do you identify an arithmetic sequence from a list of numbers?

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To identify an arithmetic sequence, check if the difference between consecutive terms is constant.

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What is the sum of the first 5 terms of the arithmetic sequence where a_1 = 2 and d = 3?

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The sum is 35.