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

Verified step by step guidance

1. Identify the general term of the sequence given by $a{n}_{}=\; 3\; imes\; 5^n$. This means each term is 3 times 5 raised to the power of n.
2. Write expressions for the specific terms needed: $a\_1\; =\; 3\; imes\; 5^1$, $a\_2\; =\; 3\; imes\; 5^2$, $a\_3\; =\; 3\; imes\; 5^3$, $a\_4\; =\; 3\; imes\; 5^4$, and $a\_5\; =\; 3\; imes\; 5^5$.
3. Calculate each ratio by dividing the corresponding terms: $\backslash frac\{a\_2\}\{a\_3\}\; =\; \backslash frac\{3\; imes\; 5^2\}\{3\; imes\; 5^3\}$, $\backslash frac\{a\_1\}\{a\_2\}\; =\; \backslash frac\{3\; imes\; 5^1\}\{3\; imes\; 5^2\}$, $\backslash frac\{a\_4\}\{a\_3\}\; =\; \backslash frac\{3\; imes\; 5^4\}\{3\; imes\; 5^3\}$, and $\backslash frac\{a\_5\}\{a\_4\}\; =\; \backslash frac\{3\; imes\; 5^5\}\{3\; imes\; 5^4\}$.
4. Simplify each ratio by canceling the common factor 3 and applying the properties of exponents: $\backslash frac\{5^m\}\{5^n\}\; =\; 5^\{m-n\}$. For example, $\backslash frac\{a\_2\}\{a\_3\}\; =\; 5^\{2-3\}\; =\; 5^\{-1\}$.
5. Observe the simplified ratios to identify any pattern or relationship, such as whether the ratios are constant or follow a specific rule related to the powers of 5.

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific formula for its nth term. Understanding how to find individual terms using the given formula an = 3 * 5^n is essential to evaluate and compare terms in the sequence.

Ratio of Consecutive Terms

The ratio of consecutive terms in a sequence is found by dividing one term by the next or previous term. This concept helps identify patterns such as constant ratios, which indicate geometric sequences.

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a fixed constant called the common ratio. Recognizing this helps explain why ratios like a2/a3 or a5/a4 are constant in sequences defined by exponential expressions.