In this set of exercises, you will use sequences to study...

In this set of exercises, you will use sequences to study real-world problems. A sequence of square boards is made as follows. The first board has dimensions 1 inch by 1 inch, the second has dimensions 2 inches by 2 inches, the third has dimensions 3 inches by 3 inches, and so on. (a) What type of sequence is formed by the perimeters of the boards? Explain. (b) Write a rule for the sequence formed by the areas of the boards. Is the sequence arithmetic, geometric, or neither? Explain your answer.

In this set of exercises, you will use sequences to study real-world problems.
A sequence of square boards is made as follows. The first board has dimensions 1 inch by 1 inch, the second has dimensions 2 inches by 2 inches, the third has dimensions 3 inches by 3 inches, and so on.
(a) What type of sequence is formed by the perimeters of the boards? Explain.
(b) Write a rule for the sequence formed by the areas of the boards. Is the sequence arithmetic, geometric, or neither? Explain your answer.

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Key Concepts

Geometric Sequence

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. This type of sequence is characterized by its multiplicative pattern. Geometric sequences commonly appear in scenarios involving exponential growth or decay.

Quadratic Sequence

A quadratic sequence is one in which the pattern of differences between consecutive terms itself forms an arithmetic sequence. Such sequences are often defined by a quadratic function of the term number. This type of sequence can model situations where the rate of change itself changes linearly, leading to a non-constant difference between consecutive terms and ruling out both arithmetic and geometric categorization.

Arithmetic Sequence

An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This constant, often called the common difference, is added to each term to get the next one. In real-world contexts, when a quantity increases by the same fixed amount at each step, the resulting sequence is arithmetic.

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