Problem 13 - College Algebra

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 are \( \frac{1}{3}, \frac{4}{5}, \frac{9}{7}, \frac{16}{9} \).

Step 1: Understanding the Formula

The formula for the sequence is given as \( a_n = \frac{n^2}{2n+1} \), where \( n \) is the term number in the sequence. We need to find the first four terms, i.e., \( a_1 \), \( a_2 \), \( a_3 \), and \( a_4 \).

Step 2: Calculate First Term

To find \( a_1 \), substitute \( n = 1 \) into the formula: \[ a_1 = \frac{1^2}{2 \times 1 + 1} = \frac{1}{3} \]

Step 3: Calculate Second Term

To find \( a_2 \), substitute \( n = 2 \) into the formula: \[ a_2 = \frac{2^2}{2 \times 2 + 1} = \frac{4}{5} \]

Step 4: Calculate Third Term

To find \( a_3 \), substitute \( n = 3 \) into the formula: \[ a_3 = \frac{3^2}{2 \times 3 + 1} = \frac{9}{7} \]

Step 5: Calculate Fourth Term

To find \( a_4 \), substitute \( n = 4 \) into the formula: \[ a_4 = \frac{4^2}{2 \times 4 + 1} = \frac{16}{9} \]

Key Concepts

Sequence Formula

A sequence formula is an equation that allows us to find the nth term of a sequence. In our exercise, the sequence is defined by the formula \( a_n = \frac{n^2}{2n + 1} \). Understanding the sequence formula is crucial because it tells us how each term relates to the position number, \( n \).
Here, the sequence is made up of fractions where the numerator is the square of \( n \), and the denominator is a simple linear expression in \( n \).

• The numerator, \( n^2 \), indicates that each term's numerator grows quadratically.
• The denominator, \( 2n + 1 \), is linear and increases linearly with each increment in \( n \).

By understanding and applying the sequence formula, we are able to calculate any term in the sequence based on its position \( n \), helping identify patterns or piece together the sequence as needed.

Term Calculation

Term calculation involves using the sequence formula to find specific terms within the sequence. In our example, we want the first four terms. This is a step-by-step process where you replace \( n \) with 1, 2, 3, and 4 respectively.

• First Term, \( a_1 \): Substitute \( n = 1 \) into the formula: \( a_1 = \frac{1^2}{2\times1 + 1} = \frac{1}{3} \)
• Second Term, \( a_2 \): Substitute \( n = 2 \) into the formula: \( a_2 = \frac{2^2}{2\times2 + 1} = \frac{4}{5} \)
• Third Term, \( a_3 \): Substitute \( n = 3 \) into the formula: \( a_3 = \frac{3^2}{2\times3+1} = \frac{9}{7} \)
• Fourth Term, \( a_4 \): Substitute \( n = 4 \) into the formula: \( a_4 = \frac{4^2}{2\times4+1} = \frac{16}{9} \)

By substituting different values of \( n \) into the sequence formula, we can find each term individually. Understanding this process allows us to manually verify any computations and ensure understanding of the sequence’s nature.

Algebraic Expressions

Algebraic expressions play a key role in understanding and working with sequences. In the given formula \( a_n = \frac{n^2}{2n + 1} \), both the numerator and the denominator are algebraic expressions.
The numerator \( n^2 \) is a simple expression involving a square function, indicating that the term's size increases significantly as \( n \) gets larger. Meanwhile, the denominator \( 2n + 1 \) is a linear expression.

• A simple quadratic like \( n^2 \) makes calculations predictable as it dictates faster growth.
• The linear expression \( 2n + 1 \) allows us to easily predict the pattern of growth.

By understanding the behavior of these algebraic expressions, we can quickly analyze how the terms change between positions. Recognizing how these expressions interact through substitution gives us insight into the sequence's trend and how rapidly terms increase.