Problem 24 - Thomas Calculus

Question

Find a formula for the \(n\)th term of the sequence. $$ \frac{1}{25}, \frac{8}{125}, \frac{27}{625}, \frac{64}{3125}, \frac{125}{15,625}, \dots $$

Step-by-Step Solution

Answer

The nth term of the sequence is \( a_n = \frac{n^3}{5^{n+1}} \).

Step 1: Identify the Pattern

First, look at the numerators in the sequence: 1, 8, 27, 64, 125. Notice that these numbers are 1^3, 2^3, 3^3, 4^3, 5^3. Therefore, the numerator of the nth term is n^3.

Step 2: Identify the Denominator Pattern

Now examine the denominators: 25, 125, 625, 3125, 15625. These are powers of 5: 5^2, 5^3, 5^4, 5^5, 5^6. Observe that the exponent on 5 is increasing by 1 with each term, starting from 2 for the first term. Thus, for the nth term, the denominator is 5^{n+1}.

Step 3: Form the General Formula

Combine the patterns from Steps 1 and 2 to create a general formula for the nth term. The nth term of the sequence is given by \( a_n = \frac{n^3}{5^{n+1}} \).

Step 4: Verify the Formula

To confirm our formula, plug in n = 1, 2, 3, and compare results with the sequence terms: \( a_1 = \frac{1}{25} \), \( a_2 = \frac{8}{125} \), \( a_3 = \frac{27}{625} \). All match the given sequence.

Key Concepts

Numerator Pattern

The pattern of the numerators is often a first step in understanding sequences. In our sequence here, the numerator sequence is: 1, 8, 27, 64, 125. It may look random at first, but these numbers can be recognized as perfect cubes.

• 1 is the cube of 1, or \( 1^3 \)
• 8 is the cube of 2, or \( 2^3 \)
• 27 is the cube of 3, or \( 3^3 \)
• 64 is the cube of 4, or \( 4^3 \)
• 125 is the cube of 5, or \( 5^3 \)

Thus, we see a consistent pattern that each term in the numerator is the cube of its position number \( n \) in the sequence. This means that for the \( n \)-th term of our sequence, the numerator can be represented as \( n^3 \). Using cubes in sequences is common and can be linked to exponential and polynomial patterns observed in mathematics.

Denominator Pattern

Denominators also provide significant insight into the pattern of a sequence. For the sequence provided, the denominators are as follows: 25, 125, 625, 3125, 15625.Similar to numerators, denominators often follow a recognizable pattern. Here, we notice these numbers are powers of 5.

• 25 is \( 5^2 \)
• 125 is \( 5^3 \)
• 625 is \( 5^4 \)
• 3125 is \( 5^5 \)
• 15625 is \( 5^6 \)

It becomes evident that the power of 5 increases consistently by 1 with each step in the sequence. Therefore, the denominator for the \( n \)-th term can be represented as \( 5^{n+1} \). Recognizing this exponential pattern is crucial, as it helps to formulate the general term of a sequence.

General Term

Formulating the general term combines the insights gained from patterns in both the numerators and denominators. Each sequence term is a fraction with a pattern-based numerator and denominator. From the insights above:

• The numerator for the \( n \)-th term is \( n^3 \).
• The denominator for the \( n \)-th term is \( 5^{n+1} \).

Thus, the general term of the sequence is represented as:\[ a_n = \frac{n^3}{5^{n+1}} \]Providing a clear mathematical expression for sequences facilitates understanding and manipulation. This general term formula summarizes the entire sequence pattern into a neat mathematical expression, making it possible to compute any term directly.

Exponential Sequence

An exponential sequence involves terms where a constant base is raised to variable exponents, often observed in denominators or numerators as indicators of rapid growth or decay. In our sequence, the exponential pattern resides in the denominator. Each denominator term is a power of 5 that increases as \( 5^{n+1} \). Such exponential relations imply that as \( n \) increases, the denominator grows significantly larger, which influences the shock of decrease or rate of smallness in the fraction's value.Understanding this exponential growth pattern facilitates insight into how sequences progress or converge. Moreover, sequences like these are found across various applications, such as geometric growth models or interest calculations in finance. Observing exponential behavior is fundamental in aligning sequence formulas in practical scenarios.