Determine whether the sequence in each question is...

Determine whether the sequence in each question is arithmetic, geometric, or neither. Find the common difference for the arithmetic ones and the common ratio for the geometric ones. Find the common difference or ratio and the 1 Oth term for each arithmetic or geometric one as appropriate. $$3,6,12,18,21,27, \dots$$

Key Concepts

Common Ratio

The common ratio is the constant factor by which each term in a geometric sequence is multiplied to obtain the next term. It is found by dividing any term by the preceding term and is essential for distinguishing geometric sequences from other types of progressions.

Identifying Sequence Type

Determining whether a sequence is arithmetic or geometric involves calculating the differences and ratios of consecutive terms. If the difference remains constant, the sequence is arithmetic; if the ratio remains constant, it is geometric. When neither of these conditions holds, the sequence is classified as neither, requiring additional analysis or different techniques for further study.

Common Difference

The common difference is the fixed amount added (or subtracted) to each term in an arithmetic sequence to arrive at the next term. It is calculated by subtracting any term from the term that immediately follows it and is a key identification element for arithmetic sequences.

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a constant factor called the common ratio. This multiplicative structure often results in exponential growth or decay, with the nth term being the first term multiplied by the common ratio raised to (n-1).

Arithmetic Sequence

An arithmetic sequence is a list of numbers where the difference between successive terms is constant, known as the common difference. This property leads to a linear progression, and the nth term can be expressed by the formula: first term plus (n-1) times the common difference.

Sequence

A sequence is an ordered list of numbers that follow a specific pattern or rule. Recognizing the type of sequence involves examining how each term relates to its neighbors, which is fundamental in understanding patterns and progressions in mathematics.

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