Problem 3 - Contemporary Precalculus

Question

Find the first five terms of the sequence \(\left\\{a_{n}\right\\}\). $$a_{n}=\frac{1}{n^{3}}$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are 1, 0.125, 0.037, 0.015625, and 0.008.

Step 1: Find a_1

Plug in n=1 into the given formula: \(a_1 = \frac{1}{1^3} = \frac{1}{1}=1\).

Step 2: Find a_2

Plug in n=2 into the given formula: \(a_2 = \frac{1}{2^3} = \frac{1}{8}=0.125\).

Step 3: Find a_3

Plug in n=3 into the given formula: \(a_3 = \frac{1}{3^3} = \frac{1}{27}\approx 0.037\).

Step 4: Find a_4

Plug in n=4 into the given formula: \(a_4 = \frac{1}{4^3} = \frac{1}{64}=0.015625\).

Step 5: Find a_5

Plug in n=5 into the given formula: \(a_5 = \frac{1}{5^3} = \frac{1}{125}=0.008\). The first five terms of the sequence are: 1, 0.125, 0.037, 0.015625, and 0.008.

Key Concepts

Sequence Terms

In mathematical sequences, each number in the list is known as a term. Sequences often have a specific rule or formula that helps you find the value of these terms. For our exercise, the sequence is defined by the formula \(a_n = \frac{1}{n^3}\). Each term \(a_n\) is determined by substituting different values of \(n\), starting from \(n=1\) and so on. This gives us the sequence of terms. Think of sequence terms as positions in a series, much like a playlist of songs where each song corresponds to a specific spot in the list.

Formula Application

The key to mastering sequences is understanding how to apply the given formula to find individual terms. Here, the formula \(a_n = \frac{1}{n^3}\) serves as the roadmap to calculate each term in the sequence.

• For the first term, substitute \(n=1\): \(a_1 = \frac{1}{1^3} = 1\).
• For the second term, substitute \(n=2\): \(a_2 = \frac{1}{2^3} = 0.125\).
• Similarly, calculate subsequent terms by incrementing \(n\) and applying the formula.

This approach ensures that each step in finding a term is guided by the structure given by the sequence formula, making complex sequences more manageable.

Fraction Simplification

In mathematical sequences like the one presented, fraction simplification plays a crucial role. It helps in making the numbers easier to work with and understand, especially when they involve powers and division.Take the given sequence formula: \(a_n = \frac{1}{n^3}\). Here, every sequence term involves calculating the cube of \(n\) and then taking its reciprocal. Simplification involves:

• Calculating \(n^3\) accurately. For example, \(3^3 = 27\).
• Taking the reciprocal: \(\frac{1}{27}\) is the simplified fraction for \(n=3\).

Using decimal approximations helps make sense of fraction values, transforming them into more understandable terms like 0.125 or 0.037.

Arithmetic Calculations

Arithmetic calculations are foundational skills required to solve sequence problems. They involve basic operations like multiplication, division, and exponentiation.For the sequence \(a_n = \frac{1}{n^3}\), follow these steps:

• Exponentiation: Calculate \(n^3\) for each term.
• Division: Take the reciprocal of the resulting number to find the decimal value.

Consider the third term: for \(n=3\), calculate \(3^3 = 27\), then find its reciprocal \(\frac{1}{27} \approx 0.037\). Practicing these calculations solidifies understanding of sequences and enhances number sense.