Determine whether the following can be the first three terms...

Determine whether the following can be the first three terms of an arithmetic or geometric sequence, and, if so, find the common difference or common ratio and the next two terms of the sequence. (A) $-11,-16,-21, \ldots$ (B) $2,-4,8, \ldots$ (C) $1,4,9, \ldots$ (D) $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \ldots$

Determine whether the following can be the first three terms of an arithmetic or geometric sequence, and, if so, find the common difference or common ratio and the next two terms of the sequence.
(A) $-11,-16,-21, \ldots$
(B) $2,-4,8, \ldots$
(C) $1,4,9, \ldots$
(D) $\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \ldots$

Key Concepts

Sequence Verification and Extension

This key concept involves checking the initial terms of a sequence to determine if they follow either an arithmetic or geometric pattern by analyzing differences and ratios. Once the pattern is established, the sequence can be extended using the respective formula, either by adding the common difference in arithmetic sequences or by multiplying by the common ratio in geometric sequences.

Common Ratio

The common ratio is the constant factor by which consecutive terms of a geometric sequence are multiplied to obtain the next term. Recognizing and computing the common ratio is important for confirming that a sequence is geometric and for calculating subsequent terms.

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant value, known as the common difference, to the previous term. Understanding arithmetic sequences involves recognizing this consistent additive pattern and using it to compute subsequent terms or solve related problems.

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is produced by multiplying the previous term by a constant factor, known as the common ratio. This concept focuses on the multiplicative nature of the sequence, allowing us to determine later terms or analyze the series by leveraging the calculated ratio.

Common Difference

The common difference is the fixed amount that is added to each term of an arithmetic sequence to reach the next term. Identifying the common difference is crucial for verifying if a series is arithmetic and for generating further terms in the sequence.

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