This set of exercises will draw on the ideas presented in...

This set of exercises will draw on the ideas presented in this section and your general math background. You are given two terms of a sequence: $a_{1}=1$ and $a_{3}=9$ (a) Find the rule for this sequence, assuming it is arithmetic. (b) Find the rule for this sequence, assuming it is geometric. (c) Find $a_{4}, \ldots, a_{8}$ for the sequence in part (a). (d) Find $a_{4}, \ldots, a_{8}$ for the sequence in part (b). (e) Which sequence grows faster, the one in part (a) or the one in part (b)? Explain your answer.

This set of exercises will draw on the ideas presented in this section and your general math background.
You are given two terms of a sequence: $a_{1}=1$ and $a_{3}=9$
(a) Find the rule for this sequence, assuming it is arithmetic.
(b) Find the rule for this sequence, assuming it is geometric.
(c) Find $a_{4}, \ldots, a_{8}$ for the sequence in part (a).
(d) Find $a_{4}, \ldots, a_{8}$ for the sequence in part (b).
(e) Which sequence grows faster, the one in part (a) or the one in part (b)? Explain your answer.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference, often denoted by d, allows us to generate any term of the sequence using the formula a? = a? + (n - 1)d. Arithmetic sequences exhibit linear behavior, meaning they increase or decrease by a fixed amount at each step.

Geometric Sequence

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed constant known as the common ratio, denoted by r. The nth term of a geometric sequence is given by the formula a? = a? · r^(n - 1). This type of sequence exhibits exponential behavior, meaning its growth or decay rate is multiplicative, which can lead to significantly faster increases or decreases compared to arithmetic sequences.

Sequence Rule (Explicit Formula)

An explicit formula for a sequence provides a direct method to calculate any term in the sequence using its position (n), without relying on previous terms. For arithmetic sequences, the explicit formula accounts for the initial term and the common difference, while for geometric sequences, it relies on the initial term and the common ratio. This concept is fundamental for understanding and predicting the behavior of sequences without recursive computations.

Growth Rate Comparison

Comparing the growth rates of sequences involves analyzing how the terms increase or decrease over time. Arithmetic sequences grow linearly, adding a constant amount to each term, while geometric sequences grow exponentially, multiplying each term by a constant factor. As a result, even if the initial terms seem comparable, geometric sequences tend to grow much faster (or decay more rapidly) than arithmetic sequences when the common ratio exceeds one (or is less than one for decay).

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