Question

Find a rule for each sequence whose first four terms are given. Assume that the given pattern will continue. $$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \ldots$$

   Find a rule for each sequence whose first four terms are given. Assume that the given pattern will continue.
$$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \ldots$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 10, Problem 26

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Solved by Expert Linh Vu

04/12/2020

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We can see that each term is obtained by dividing the previous term by 4. Show more…

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Find a rule for each sequence whose first four terms are given. Assume that the given pattern will continue. $$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \ldots$$

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Key Concepts

Common Ratio

The common ratio in a geometric sequence is the constant value by which each term is multiplied to obtain the subsequent term. Identifying the common ratio is essential for determining the behavior of the sequence, whether it is increasing, decreasing, or remaining constant, and it serves as the basis for the formula of the sequence.

Explicit Formula for a Geometric Sequence

The explicit formula for a geometric sequence allows one to calculate any term directly without needing to compute all the previous terms. It is generally expressed as a? = a? * r^(n-1), where a? is the first term, r is the common ratio, and n is the term number. This formula is crucial for efficiently solving problems that involve finding specific terms in the sequence.

Geometric Sequence

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio. This type of sequence is characterized by its multiplicative pattern, and it is fundamental to understanding exponential growth or decay processes in various fields of mathematics.

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