Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence 1, −2, 4, −8, 16, ………. Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

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1. Identify the terms of the sequence given: 1, -2, 4, -8, 16, ... and label them as $a\_1\; =\; 1$, $a\_2\; =\; -2$, $a\_3\; =\; 4$, $a\_4\; =\; -8$, and $a\_5\; =\; 16$.
2. Calculate the ratio $\backslash frac\{a\_2\}\{a\_3\}$ by dividing the second term by the third term: $\backslash frac\{-2\}\{4\}$.
3. Calculate the ratio $\backslash frac\{a\_1\}\{a\_2\}$ by dividing the first term by the second term: $\backslash frac\{1\}\{-2\}$.
4. Calculate the ratio $\backslash frac\{a\_4\}\{a\_3\}$ by dividing the fourth term by the third term: $\backslash frac\{-8\}\{4\}$.
5. Calculate the ratio $\backslash frac\{a\_5\}\{a\_4\}$ by dividing the fifth term by the fourth term: $\backslash frac\{16\}\{-8\}$. Then, observe the pattern in these ratios to understand the behavior of the sequence.

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term, denoted as a₁, a₂, a₃, etc. Understanding how to identify and refer to terms is essential for analyzing relationships between them.

Ratio of Terms in a Sequence

The ratio between terms, such as a₂/a₃, compares the values of two terms in the sequence. Calculating these ratios helps identify patterns, especially in geometric sequences where the ratio between consecutive terms is constant.

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a fixed number called the common ratio. Recognizing this pattern allows for predicting terms and understanding the behavior of the sequence.