Problem 9 - College Algebra

Question

Write the first four terms of the sequence. $$a_{n}=\frac{2^{n}}{n^{3}}$$

Step-by-Step Solution

Answer

The first four terms are: 2, 1/2, 8/27, and 1/4.

Step 1: Understanding the Sequence Formula

The given formula for the sequence is \( a_n = \frac{2^n}{n^3} \). This formula describes each term of the sequence as a fraction where the numerator is \( 2^n \) and the denominator is \( n^3 \). Our goal is to substitute the first four natural numbers for \( n \) to find the corresponding terms of the sequence.

Step 2: Calculating the First Term

Substitute \( n = 1 \) into the formula: \( a_1 = \frac{2^1}{1^3} = \frac{2}{1} = 2 \). Therefore, the first term of the sequence is 2.

Step 3: Calculating the Second Term

Substitute \( n = 2 \) into the formula: \( a_2 = \frac{2^2}{2^3} = \frac{4}{8} = \frac{1}{2} \). Thus, the second term of the sequence is \( \frac{1}{2} \).

Step 4: Calculating the Third Term

Substitute \( n = 3 \) into the formula: \( a_3 = \frac{2^3}{3^3} = \frac{8}{27} \). Therefore, the third term of the sequence is \( \frac{8}{27} \).

Step 5: Calculating the Fourth Term

Substitute \( n = 4 \) into the formula: \( a_4 = \frac{2^4}{4^3} = \frac{16}{64} = \frac{1}{4} \). Thus, the fourth term of the sequence is \( \frac{1}{4} \).

Key Concepts

Sequence Formula

A sequence in mathematics is essentially a list of numbers that follow a specific pattern or rule. The sequence formula provides the means to determine each term in this list. In our case, the sequence formula is given by \[a_n = \frac{2^n}{n^3}\]. This particular formula lays out a clear structure for each term in the sequence, where:

• The numerator, \(2^n\), represents 2 raised to the nth power.
• The denominator, \(n^3\), represents n raised to the third power.

By substituting different values of \(n\), we can generate the terms of the sequence. This automatic generation is what makes sequence formulas powerful tools in mathematics. Each term is calculated independently by applying the same rule set by the sequence formula.

Natural Numbers

Natural numbers are a fundamental part of mathematics. They are the numbers we naturally count with, starting from 1 and going upwards without limit.

• Examples of natural numbers are 1, 2, 3, 4, and so on.
• Natural numbers are positive and do not include fractions, decimals, or negative numbers.

In the context of sequences, natural numbers are often used as indices. This means that when we're looking at a sequence formula like \(a_n = \frac{2^n}{n^3}\), the \(n\) in the formula will take on values from the set of natural numbers: 1, 2, 3, etc. This allows us to systematically calculate each term in order, starting from the first term, all the way to the nth term.

Fraction Terms

Understanding fractions is key when working with sequences involving fraction terms. A fraction represents a part of a whole and is composed of two parts: the numerator and the denominator. In the sequence formula \(a_n = \frac{2^n}{n^3}\), each term is expressed as a fraction:

• The numerator \(2^n\) grows exponentially as \(n\) increases.
• The denominator \(n^3\) grows more rapidly because it's a cubic function of \(n\).
• The fraction as a whole, \(\frac{2^n}{n^3}\), diminishes because the denominator typically increases faster than the numerator.

This pattern leads to interesting sequences, where the value of each term might shrink, even while the individual components grow. This interplay between numerator and denominator determines the behavior and progression of the sequence over various values of \(n\).