Question

A sequence is given by \(5, \frac{5}{8}, \frac{5}{27}, \frac{5}{64}, \ldots\) Write down an expression to denote the full sequence.

Short answer

Answer: The general expression for the given sequence is \(\frac{5}{n^3}\), where n is the position of the term in the sequence.

Step-by-step solution

Identify the pattern in the sequence

Observe the given sequence: \(5, \frac{5}{8}, \frac{5}{27}, \frac{5}{64}, \ldots\). Notice that each term's numerator is 5, and the denominators are increasing in powers of 2, 3, 4, and so on. Therefore, the sequence can be rewritten as: \(5, \frac{5}{2^3}, \frac{5}{3^3}, \frac{5}{4^3}, \ldots\).

Find the general expression for the sequence

From the pattern identified in Step 1, we can determine the general expression for the nth term using the following format: \(\frac{5}{n^3}\), where n is the position of the term in the sequence.

Write down the expression for the full sequence

Using the general expression from Step 2, we can write down the expression to denote the full sequence as: \(\left\{\frac{5}{n^3}\right\}_{n=1}^{\infty}\). This means the sequence consists of terms where the numerator is 5 and the denominator is the cube of each natural number n, starting from 1 to infinity.

Key concepts

Sequence Pattern Recognition

Recognizing patterns in a sequence is crucial in mathematics. This ability to identify recurring or distinguishing elements can help decipher the sequence's rule. In the exercise provided, the pattern is observed in the denominators of the sequence:

• First term: 5 (denominator 1)
• Second term: \( \frac{5}{8} \) (denominator \(2^3\))
• Third term: \( \frac{5}{27} \) (denominator \(3^3\))
• Fourth term: \( \frac{5}{64} \) (denominator \(4^3\))

Each term's denominator is a consecutive integer raised to the power of three. This tells us that the pattern in the denominator involves increasing powers of the sequence index (1, 2, 3,...) cubed.
Understanding this pattern lets us express the sequence systematically. Each term is then constructed by taking a constant numerator and varying the denominator according to this identified pattern. Recognizing such patterns is foundational to understanding both simple and complex sequences.

General Term of a Sequence

A general term is a formula that defines the nth term of a sequence without writing out all previous elements. In the exercise, the general term was identified as \( \frac{5}{n^3} \).
Here, the term 'n' represents the position of each term in the sequence.

• The numerator remains constant at 5 for all terms.
• The denominator is the cube of the term’s position in the sequence.

To find any term within the sequence, substitute the desired position number for 'n' in the general term formula \( \frac{5}{n^3} \).
This provides a systematic means to predict any term's value without listing every term.
Such expressions are powerful, clarifying the rule governing the sequence and simplifying calculations for terms located further within the sequence.

Infinite Sequences

Infinite sequences continue indefinitely, having no final term. In the original problem, the sequence is expressed as \(\left\{\frac{5}{n^3}\right\}_{n=1}^{\infty}\). This notation indicates the sequence contains an unending number of terms.
Representing infinite sequences includes writing a rule or general expression valid as 'n' increases without bound. The sequence's general expression, \( \frac{5}{n^3} \), clarifies this unbounded nature by showing how each term must be calculated.
Infinite sequences are fundamental in mathematics as they illustrate continually progressing scenarios, found frequently in calculus and other areas.

• They require careful manipulation due to their endless nature.
• Understanding convergence or divergence of sequences is often significant in further studies.

Mastering the concept of infinity in sequences can deepen mathematical insight, leading to a greater grasp of continuous mathematical processes.