6.3 Arithmetic & Geometric Sequences and Series

Front

What is an arithmetic sequence?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d).

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How do you find the nth term of an arithmetic sequence?

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The nth term (a_n) 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.

Front

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 (r).

Front

How do you find 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.

Front

What is the formula for the sum of the first n terms of an arithmetic series?

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The sum (S_n) of the first n terms of an arithmetic series 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.

Front

What is the formula for the sum of the first n terms of a 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), for r ≠ 1.

Front

If a_15 = 108 and d = 7, what is a_1?

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a_1 = 10.

Front

Find the sum of the geometric series where a_1 = 4, r = 2, and a_n = 16,384. What is the sum?

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The sum is 32,764.

Front

What are the next two terms of the sequence 8, 4, 2, 1, 1/2?

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The next two terms are 1/4 and 1/8.

Front

What is the common difference in the arithmetic sequence 2, 17, 32, 47, ...?

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The common difference (d) is 15.