Problem 9 - Algebra and Trigonometry

Question

write the first four terms of each sequence whose general term is given. $$ a_{n}=\frac{2 n}{n+4} $$

Step-by-Step Solution

Answer

The first four terms of the sequence are \(a_{1}=\frac{2}{5}\), \(a_{2}=\frac{2}{3}\), \(a_{3}=\frac{6}{7}\), \(a_{4}=1\)

Step 1: Substitute \(n=1\) in the formula and calculate \(a_{1}\)

Substitute \(n=1\) into the formula \(a_{n}=\frac{2n}{n+4}\), which gives \(a_{1}=\frac{2(1)}{1+4}= \frac{2}{5}\)

Step 2: Substitute \(n=2\) in the formula and calculate \(a_{2}\)

Substitute \(n=2\) into the formula \(a_{n}=\frac{2n}{n+4}\), which gives \(a_{2}=\frac{2(2)}{2+4}= \frac{4}{6}\). To simplify this, \(a_{2}=\frac{2}{3}\)

Step 3: Substitute \(n=3\) in the formula and calculate \(a_{3}\)

Substitute \(n=3\) into the formula \(a_{n}=\frac{2n}{n+4}\), which gives \(a_{3}=\frac{2(3)}{3+4}= \frac{6}{7}\)

Step 4: Substitute \(n=4\) in the formula and calculate \(a_{4}\)

Substitute \(n=4\) into the formula \(a_{n}=\frac{2n}{n+4}\), which gives \(a_{4}=\frac{2(4)}{4+4}= \frac{8}{8}\). To simplify this, \(a_{4}=1\)

Key Concepts

General Term

Understanding the concept of a general term is crucial when working with sequences. A sequence essentially is a list of numbers following a certain rule. The general term provides us with a formula to generate any term within that sequence.
For example, the general term given in the exercise is \(a_{n}=\frac{2n}{n+4}\).
This means that for any term in the sequence, we can substitute a value of \(n\) to find the respective term value.

• "\(n\)" represents the position of the term in the sequence. If you want the first term, you substitute \(n=1\).
• The general term allows prediction of patterns or properties of a sequence without listing all terms.

This formulaic approach offers a clear and systematic way to find terms irrespective of the sequence length.

Substitution

Substitution is the process where we replace a variable with a specific value. In sequences, we substitute different numbers for \(n\) to get the sequence's terms.
Through substitution, you directly apply the general term formula for different positions in the sequence. For example, to find \(a_{1}\), we substitute \(n=1\) in the expression \(a_{n}=\frac{2n}{n+4}\).

• This substitution gives us \(a_{1}=\frac{2(1)}{1+4}=\frac{2}{5}\).
• Similarly, substituting \(n=2, 3,\) and \(4\) yields \(a_{2}=\frac{2}{3},\) \(a_{3}=\frac{6}{7},\) and \(a_{4}=1\) respectively.

Substitution is a straightforward yet powerful tool to derive specific values from a formula.

Simplification

Simplification makes mathematical expressions easier to understand and work with. In the context of sequences, after substitution, you might end up with fractions that can be reduced.
Take, for instance, when \(n=2\), substituting gives \(a_{2}=\frac{4}{6}\).
To simplify \(\frac{4}{6}\), we divide both the numerator and denominator by their greatest common divisor, which is \(2\).
This results in \(\frac{2}{3}\), a more simplified expression for \(a_{2}\).

• Simplification helps in revealing properties of the sequence, like identifying repeating patterns.
• It makes calculations less cumbersome, especially when dealing with longer sequences.

Simplifying terms makes your work more understandable and organized.

Mathematical Formula

A mathematical formula provides a concise way of expressing abstract concepts. In sequences, formulas capture the essence of the sequence in a single expression.
The formula \(a_{n}=\frac{2n}{n+4}\) provides a complete vision of how to generate terms. It captures both the numerator \(2n\) (representing the growth at each step) and the denominator \(n+4\) (showing how the position \(n\) interacts with the term).

• Formulas enable analysis of data patterns, helping derive other formulas or insights.
• They offer a framework to extend knowledge and solve more complex problems.

Such mathematical formulas are foundational in many branches of math, ensuring uniformity and simplicity in handling data.