Geometric Sequences

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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 formula for the nth term of a geometric sequence?

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The nth term (a_n) of a geometric sequence can be calculated 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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How do you find the sum of a finite geometric series?

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The sum (S_n) of the first n terms of a geometric series can be calculated using the formula: 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 the formula for the sum of an infinite geometric series?

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The sum (S) of an infinite geometric series exists if the absolute value of the common ratio |r| < 1, and is given by the formula: S = a_1 / (1 - r), where a_1 is the first term.

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What is the common ratio in a geometric sequence?

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

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The sum is 1.

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If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 5th term?

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The 5th term is 48.

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What happens to the sum of an infinite geometric series if the common ratio is greater than or equal to 1?

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If the common ratio is greater than or equal to 1, the sum of the infinite geometric series does not exist.

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Find the 11th term of the geometric sequence: 64, -32, 16, -8, ...

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The 11th term is -0.0625.

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The sum is 5.