13. Arithmetic Sequences Explicit to Linear Functions

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What is an arithmetic sequence?

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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 nth term of an arithmetic sequence?

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

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What is the explicit formula for an arithmetic sequence?

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The explicit formula for an arithmetic sequence is a_n = a_1 + (n-1)d.

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What is a linear function?

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A linear function is a function that can be graphically represented as a straight line. It has the form f(x) = mx + b, where m is the slope and b is the y-intercept.

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How can you convert an arithmetic sequence to a linear function?

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You can convert an arithmetic sequence to a linear function by expressing the nth term of the sequence as a function of n, typically in the form f(n) = a_1 + (n-1)d.

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What is the common difference in an arithmetic sequence?

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The common difference is the constant amount that each term in the sequence increases or decreases by. It is calculated as d = a_n - a_(n-1).

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What does it mean to rewrite an explicit formula in function form?

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Rewriting an explicit formula in function form means expressing the formula in terms of a function notation, typically f(n) instead of a_n.

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Give an example of an arithmetic sequence.

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An example of an arithmetic sequence is 2, 5, 8, 11, 14, ... where the common difference is 3.

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What is the first term of an arithmetic sequence?

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The first term of an arithmetic sequence is denoted as a_1 and is the starting point of the sequence.

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What is the slope in a linear function?

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The slope (m) in a linear function represents the rate of change of the function. It indicates how much the output value (y) changes for a unit change in the input value (x).