Sequences and Series Review

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What is an arithmetic sequence?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. For example, 4, 6, 8, 10 is an arithmetic sequence with a common difference of 2.

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Identify the common difference in the arithmetic sequence: 4, 1, -2, -5, -8.

Back

The common difference is -3, as each term is obtained by subtracting 3 from the previous term.

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

Back

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

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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. For example, 2, 4, 8, 16 is a geometric sequence with a common ratio of 2.

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What is the formula for the nth term of a geometric sequence?

Back

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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Find the 1st term of the geometric sequence defined by the rule: a_n = 10(2^(n-1)).

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The 1st term (a_1) is 10, as substituting n=1 gives a_1 = 10(2^(1-1)) = 10.

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

Back

The sum of the first n terms (S_n) of a geometric sequence 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, provided r ≠ 1.

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Calculate the sum of the first 100 terms of the sequence: 100, 98, 96, ...

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The sum is 100, as it is an arithmetic sequence with a common difference of -2.

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

Back

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.

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Find the 22nd term of the arithmetic sequence: 5, 8, 11, ...

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The 22nd term is 68, calculated using the formula a_n = 5 + (22-1)3.