Problem 24 - Beginning and Intermediate Algebra

Question

Find a formula for the general term, \(a_{n},\) of each sequence. $$1,8,27,64, \dots$$

Step-by-Step Solution

Answer

The formula for the general term, \(a_n,\) of the sequence is: \(a_n = n^3\).

Step 1: Identify the pattern

Observe the given sequence and identify the pattern. Notice that the sequence can be represented as: $$1^3, 2^3, 3^3, 4^3, \dots$$

Step 2: Write down the general term

Since the sequence is a list of cubes of consecutive natural numbers, the general term, \(a_{n},\) can be written as: $$a_n = n^3$$

Step 3: Verify with given terms

Verify that this general term formula holds for the given sequence: \(a_1 = 1^3 = 1\) \(a_2 = 2^3 = 8\) \(a_3 = 3^3 = 27\) \(a_4 = 4^3 = 64\) The formula \(a_n = n^3\) is indeed valid for the given terms. Thus, the formula for the general term, \(a_n,\) of the sequence is: $$ a_n = n^3 $$

Key Concepts

Cube Numbers

A cube number is the result of multiplying a number by itself three times. This means if you have a number \( n \), the cube of it is expressed as \( n^3 \). For example, if \( n = 2 \), then \( n^3 = 2 \times 2 \times 2 = 8 \). Cube numbers come from a specific pattern where every term is the cube of a natural number.
They grow rapidly because you are effectively multiplying more with each step compared to square numbers, which makes them very distinct.
Understanding cube numbers is important when working with mathematical sequences, as they often form nice, predictable patterns.

• 1³ is 1 because 1×1×1 equals 1.
• 2³ is 8 because 2×2×2 equals 8.
• 3³ is 27 because 3×3×3 equals 27.
• 4³ is 64 because 4×4×4 equals 64.

This predictable pattern makes them easier to identify and work with in sequences.

General Term

In a mathematical sequence, the general term is a formula that lets you find any term in the sequence by substituting the position number into the formula.
Understanding how to express the general term allows you to easily extend the sequence without having to manually calculate each term.
For the sequence \(1, 8, 27, 64, \dots\), we notice that it consists of cube numbers, which leads us to the formula for the general term: \(a_n = n^3\).
This formula states that the term in the \(n\)-th position of the sequence is obtained by cubing \(n\). Here's how this works:

• For \(a_1\), putting \(1\) gives \(1^3 = 1\).
• For \(a_2\), putting \(2\) results in \(2^3 = 8\).
• For \(a_3\), substituting \(3\) gives \(3^3 = 27\).
• For \(a_4\), substituting \(4\) results in \(4^3 = 64\).

This pattern makes predicting and calculating further elements of the sequence straightforward and efficient.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers arranged according to a definite rule or pattern.
They are fundamental in mathematics because they help us understand patterns, predict future terms, and solve various types of problems.
The sequence given: \(1, 8, 27, 64, \dots\), is a perfect example of a sequence generated using cube numbers, where each term follows from the last according to the rule \(n^3\).
Sequences can be finite, like the first few terms before a rule changes, or infinite, continuing indefinitely as in this example.

• Each number in a sequence is called a term.
• The position of a term is usually denoted by \(n\).
• The rule defining a sequence is critical to finding any term.

Being able to spot the rule, as shown with cube numbers here, helps in both academic exercises and real-world applications in science and engineering.