Exercises 88–90 will help you prepare for the material covered in the next section. Consider the sequence whose nth term is an = (3)5^n Find a2/a3, a1/a2, a4/a3 and a5/a4 What do you observe?

Verified step by step guidance

1. Identify the general formula for the nth term of the sequence: $a_n = 3 \cdot 5^n$.
2. Calculate $a_2$ by substituting $n = 2$ into the formula: $a_2 = 3 \cdot 5^2$.
3. Calculate $a_3$ by substituting $n = 3$ into the formula: $a_3 = 3 \cdot 5^3$.
4. Find the ratio $\frac{a_2}{a_3}$ by dividing $a_2$ by $a_3$.
5. Repeat the process for $\frac{a_1}{a_2}$, $\frac{a_4}{a_3}$, and $\frac{a_5}{a_4}$ by calculating each term and then dividing the respective terms.

Key Concepts

Sequences

A sequence is an ordered list of numbers defined by a specific rule or formula. In this case, the nth term of the sequence is given by a_n = 3 * 5^n, where n is a non-negative integer. Understanding how to compute terms in a sequence is essential for evaluating ratios like a2/a3 and others.

Ratios

A ratio is a comparison of two quantities, showing how many times one value contains or is contained within the other. In this exercise, you will calculate ratios of consecutive terms in the sequence, such as a2/a3, which helps in analyzing the behavior and relationships between the terms.

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, often represented by functions of the form a_n = a * b^n. In this sequence, the term 5^n indicates exponential growth, which is crucial for understanding how the terms behave as n increases and for making observations about the ratios.