Arithmetic/Geometric Sequences & Series

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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, known as the common difference.

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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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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 and d is the common difference.

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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 an arithmetic series?

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The sum of the first n terms of an arithmetic series is given by: S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

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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 of the first n terms of a geometric series is given by: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.

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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 increases or decreases by in an arithmetic sequence.

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What is the common ratio in a geometric sequence?

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The common ratio is the fixed number that each term is multiplied by to get the next term in a geometric sequence.

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Evaluate the 5th partial sum of the geometric sequence: 2, 12, 72, ...

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The 5th partial sum is 3110.

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List the first five terms of the geometric sequence starting with 2 and a common ratio of 1/4.

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The first five terms are: 2, 1/2, 1/8, 1/32, 1/128.