Sequences, series, and probability

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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What is the formula for the sum of an infinite geometric series?

Back

The sum S of an infinite geometric series can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio (|r| < 1).

Front

What is the common ratio in the geometric sequence: 512, 256, 128, ...?

Back

The common ratio is 1/2, as each term is half of the previous term.

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How do you find the sum of a finite geometric series?

Back

The sum S of a finite geometric series can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.

Front

What is a common difference in an arithmetic sequence?

Back

The common difference is the constant amount that each term in an arithmetic sequence increases or decreases by, calculated as the difference between any two consecutive terms.

Front

Identify the common difference in the sequence: 97, 86, 75, 64, ...

Back

The common difference is -11.

Front

What is the explicit formula for an arithmetic sequence?

Back

The explicit formula for an arithmetic sequence is given by a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number.

Front

What is the explicit equation for the sequence: 25, 21, 17, 13,...?

Back

The explicit equation is a_n = -4n + 29.

Front

How do you calculate the number of combinations of options?

Back

To calculate the number of combinations, multiply the number of choices for each option together. For example, if there are 4 trim levels, 5 colors, and 3 interiors, the total combinations are 4 * 5 * 3 = 60.

Front

What is the difference between a sequence and a series?

Back

A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.