Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, 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, \(\frac{1}{2}\), \(\frac{8}{27}\), and \(\frac{1}{4}\).

Step 1: Understand the Formula for the Sequence

The sequence is given by \( a_n = \frac{2^n}{n^3} \). This means that for each term, \( n \), you substitute into the formula where \( 2^n \) is the number 2 raised to the power of \( n \), and \( n^3 \) is \( n \) raised to the power of 3.

Step 2: Calculate the First Term \(a_1\)

Substitute \( n = 1 \) into the formula: \[ a_1 = \frac{2^1}{1^3} = \frac{2}{1} = 2 \].

Step 3: Calculate the Second Term \(a_2\)

Substitute \( n = 2 \) into the formula: \[ a_2 = \frac{2^2}{2^3} = \frac{4}{8} = \frac{1}{2} \].

Step 4: Calculate the Third Term \(a_3\)

Substitute \( n = 3 \) into the formula: \[ a_3 = \frac{2^3}{3^3} = \frac{8}{27} \].

Step 5: Calculate the Fourth Term \(a_4\)

Substitute \( n = 4 \) into the formula: \[ a_4 = \frac{2^4}{4^3} = \frac{16}{64} = \frac{1}{4} \].

Key Concepts

Exponential Functions

Exponential functions play a critical role in mathematics, particularly when expressing quantities that grow or diminish at constant rates. An exponential function looks like this: \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base of the exponent, and \( x \) is the exponent or power.
The base \( b \) dictates how quickly the function grows; a base greater than 1 leads to exponential growth, while a base between 0 and 1 results in exponential decay.
In our sequence, \( 2^n \) is an exponential term where 2 is the base, and \( n \) is the exponent. This means each term grows exponentially as \( n \) increases.

• Improving problem-solving skills often starts with understanding how to manipulate such exponential expressions.
• It's essential to practice calculating the power to get comfortable with exponential concepts.

Power of a Number

The power of a number refers to how many times that number is multiplied by itself. This is a fundamental concept in sequences and series. To raise a number \( a \) to the \( n \)th power is mathematically noted as \( a^n \).
Knowing the power of a number helps significantly in comprehending formulas like \( 2^n \) in the exercise.
Raising to a power follows these simple rules:

• \( a^1 = a \)
• \( a^0 = 1 \) for any non-zero \( a \)
• \( a^{-n} = \frac{1}{a^n} \)

These rules allow us to simplify expressions efficiently, paving the way for easier calculation of terms within sequences.
Concepts like these are broadly applicable, whether you're calculating terms in a sequence or solving equations in algebra.

Fractions in Algebra

Fractions form a key component in algebra, acting as a bridge between various numerical expressions. They express a division operation, such as \( \frac{a}{b} \), which mathematically translates to \( a \div b \).
In sequences like \( a_n = \frac{2^n}{n^3} \), fractions link exponential expressions and polynomial terms.
When simplifying such expressions:

• Simplify the numerator and denominator separately first if possible.
• Factor and reduce by canceling common factors between numerator and denominator.

Understanding fractions also includes knowing how to manipulate them in equations, such as finding common denominators or converting improper fractions.
By mastering fractions, students enhance their ability to tackle complex variables and equations in mathematics.