Geometric Sequences

Front

What is a geometric sequence?

Back

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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How do you determine if a sequence is geometric?

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To determine if a sequence is geometric, check if the ratio of consecutive terms is constant.

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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 each term in a geometric sequence is multiplied to obtain the next term.

Front

If the first term of a geometric sequence is 3 and the common ratio is 2, what are the first five terms?

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The first five terms are 3, 6, 12, 24, 48.

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What is the formula for 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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Can a geometric sequence have a common ratio of zero?

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No, a geometric sequence cannot have a common ratio of zero, as this would result in all terms being zero after the first term.

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What is the relationship between geometric sequences and exponential functions?

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Geometric sequences can be represented by exponential functions, as each term can be expressed as a constant multiplied by a power of the common ratio.

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How do you find the common ratio from a geometric sequence?

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To find the common ratio, divide any term by the previous term (r = a_n / a_(n-1)).

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What is an example of a geometric sequence?

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An example of a geometric sequence is 2, 6, 18, 54, ... with a common ratio of 3.

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What is the sum of the first n terms of a geometric sequence?

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The sum of the first n terms of a geometric sequence can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where r is not equal to 1.