Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n^{2}}{2 n+1} $$

Step-by-Step Solution

Answer

The first four terms of the sequence are \( \frac{1}{3} \), \( \frac{4}{5} \), \( \frac{9}{7} \), and \( \frac{16}{9} \).

Step 1: Identify with n=1

To find the first term of the sequence, set \( n = 1 \) and substitute it into the formula \( a_n = \frac{n^2}{2n+1} \). We get \( a_1 = \frac{1^2}{2(1) + 1} = \frac{1}{3} \).

Step 2: Compute for n=2

Now, set \( n = 2 \) to find the second term. Substitute into the formula: \( a_2 = \frac{2^2}{2(2) + 1} = \frac{4}{5} \).

Step 3: Calculate for n=3

For the third term of the sequence, let \( n = 3 \). Substitute to find: \( a_3 = \frac{3^2}{2(3) + 1} = \frac{9}{7} \).

Step 4: Evaluate for n=4

For the fourth term, set \( n = 4 \) and substitute into the sequence formula: \( a_4 = \frac{4^2}{2(4) + 1} = \frac{16}{9} \).

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a type of sequence where each term is found by adding a constant value—called the "common difference"—to the previous term. Unlike an arithmetic sequence, in our exercise, each term of the sequence is not generated by adding a constant but rather by using a specific formula.

In classic arithmetic sequences, the formula to find the nth term is:

• \( a_n = a_1 + (n-1)d \)

where

• \( a_1 \) is the first term of the sequence,
• \( d \) is the common difference,
• and \( n \) is the term number.

In contrast, the sequence from the original exercise uses a different formula specific to each term. Remember:

• The original formula representation provides a functional view to understand how other types of sequences work beyond arithmetic sequences.

Algebraic Expressions

Algebraic expressions play a crucial role in forming and solving sequences. They consist of numbers, variables, and arithmetic operations. In our exercise, the sequence is represented by the algebraic expression \( a_n = \frac{n^2}{2n+1} \).

This expression combines the variables \( n \) and several arithmetic operations:

• Squares of \( n \)
• Multiplication
• Addition
• Division

Each operation contributes to transforming the input \( n \) into a sequence term \( a_n \). Recognize

By plugging in different values of \( n \), as shown in the solution steps, the formula gives unique outcomes for each sequence term. Understanding each component of the expression and how they interact is vital for mastering sequences in algebra.

Problem-Solving in Algebra

Problem-solving in algebra often involves recognizing patterns and applying formulas accurately. Sequences are an excellent way to practice these skills. Our specific exercise involves evaluating a formula at successive integer values.

To solve such problems:

• Start by understanding the formula—know what \( n \) represents and how it affects the term \( a_n \).
• Substitute values for \( n \) into the formula step-by-step, a critical skill in problem-solving. Each term depends on replacing \( n \) with integers like 1, 2, 3, and so on.
• Calculate each part carefully. For instance, compute \( n^2 \) before performing division or addition, following the order of operations.
• Finally, understand sequences as a whole. Note how each calculated term fits into the broader pattern defined by the sequence formula.

This methodical approach helps ensure accuracy and comprehension, laying a solid foundation for more complex algebraic problem-solving.