Problem 5 - Precalculus

Question

Write the first four terms of each sequence. Assume \(n\) starts at 1. $$a_{n}=\frac{n}{(n+1)}$$

Step-by-Step Solution

Answer

The first four terms are \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}\).

Step 1: Substitute n = 1

To find the first term of the sequence, substitute \(n = 1\) in the expression \(a_n = \frac{n}{n+1}\). This gives us: \[ a_1 = \frac{1}{1+1} = \frac{1}{2} \].

Step 2: Substitute n = 2

To find the second term of the sequence, substitute \(n = 2\) in the expression \(a_n = \frac{n}{n+1}\). This gives us: \[ a_2 = \frac{2}{2+1} = \frac{2}{3} \].

Step 3: Substitute n = 3

To find the third term of the sequence, substitute \(n = 3\) in the expression \(a_n = \frac{n}{n+1}\). This gives us: \[ a_3 = \frac{3}{3+1} = \frac{3}{4} \].

Step 4: Substitute n = 4

To find the fourth term of the sequence, substitute \(n = 4\) in the expression \(a_n = \frac{n}{n+1}\). This gives us: \[ a_4 = \frac{4}{4+1} = \frac{4}{5} \].

Key Concepts

Arithmetic sequences

Arithmetic sequences are sequences of numbers where the difference between consecutive terms is constant. This difference is known as the "common difference" and is usually denoted by the letter \(d\). For an arithmetic sequence, if you know the first term \(a_1\) and the common difference \(d\), you can find any term in the sequence using the formula:

• First term: \( a_1 \)
• Common difference: \( d \)
• General term formula: \( a_n = a_1 + (n-1) \times d \)

This type of sequence is "linear" because the difference between each pair of terms is always the same.
Understanding arithmetic sequences is crucial because they form the foundation of many mathematical concepts and have practical applications in fields such as finance, computer science, and physics. Using the formula you can easily compute any term in the sequence, which can be valuable for identifying patterns and making predictions.

Sequence terms

In any sequence, a term represents an individual element or number in that series. Understanding sequence terms is very important, especially when you're studying how sequences work.
In this particular exercise, we are given a formula to find the terms of the sequence: \[ a_n = \frac{n}{n+1} \] Here, \(a_n\) represents the \(n\)-th term in the sequence. To find any term, you simply substitute the desired term number for \(n\) in the formula.

• Example 1st term: \( a_1 = \frac{1}{1+1} = \frac{1}{2} \)
• Example 2nd term: \( a_2 = \frac{2}{2+1} = \frac{2}{3} \)
• Example 3rd term: \( a_3 = \frac{3}{3+1} = \frac{3}{4} \)
• Example 4th term: \( a_4 = \frac{4}{4+1} = \frac{4}{5} \)

Being able to find specific sequence terms allows you to identify patterns, understand the progress of a sequence, and solve related problems.

Mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operators that represent a value. They serve as the foundation of algebraic thinking and problem-solving.
In a mathematical expression like \( a_n = \frac{n}{n+1} \), it's important to understand how each part contributes to the calculation:

• \(n\) is a variable representing the position of the term in the sequence.
• \(\frac{n}{n+1}\) is the expression that defines how each term is calculated.

Substituting actual values for \(n\) yields specific terms in the sequence. This exercise demonstrates the application of mathematical expressions by assigning values to a variable \(n\) to calculate and generate terms of a sequence.
The concept of mathematical expressions extends beyond sequences, acting as a fundamental tool in equations, functions, and algebraic modeling. It encourages logical thinking and helps students develop the ability to create and manipulate formulas, a skill valuable in advanced math fields.