Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{2 n+1}{n^{3}} $$

Step-by-Step Solution

Answer

The first four terms of the sequence are 3, \( \frac{5}{8} \), \( \frac{7}{27} \), and \( \frac{9}{64} \).

Step 1: Understand the Sequence Formula

The sequence is given by the formula \( a_{n} = \frac{2n+1}{n^3} \). This means each term \( a_{n} \) is calculated by substituting \( n \) with the term number in the formula. To find the first four terms, we need to substitute \( n \) with the values 1, 2, 3, and 4, respectively.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula: \[ a_1 = \frac{2(1) + 1}{1^3} = \frac{2 + 1}{1} = \frac{3}{1} = 3 \]. So, the first term \( a_1 \) is 3.

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula: \[ a_2 = \frac{2(2) + 1}{2^3} = \frac{4 + 1}{8} = \frac{5}{8} \]. Thus, the second term \( a_2 \) is \( \frac{5}{8} \).

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula: \[ a_3 = \frac{2(3) + 1}{3^3} = \frac{6 + 1}{27} = \frac{7}{27} \]. Hence, the third term \( a_3 \) is \( \frac{7}{27} \).

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula: \[ a_4 = \frac{2(4) + 1}{4^3} = \frac{8 + 1}{64} = \frac{9}{64} \]. Thus, the fourth term \( a_4 \) is \( \frac{9}{64} \).

Key Concepts

Arithmetic sequences

An arithmetic sequence is a type of mathematical sequence where the difference between consecutive terms is constant. This difference, known as the common difference, remains the same throughout the sequence. Arithmetic sequences are a straightforward example of sequences and provide a basis for understanding more complex series.

In arithmetic sequences, each term can be calculated using the formula:

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

For example, consider the sequence generated by adding 3 each time: 1, 4, 7, 10, ... This shows a constant rate of increase, which is a hallmark of arithmetic sequences.

Recognizing arithmetic sequences is all about spotting the consistent pattern. Each term grows by exactly the same amount as the previous one, which makes them easy to predict and analyze.

Mathematical sequences

Mathematical sequences refer to ordered lists of numbers and can be arithmetic or of other forms, like geometric or even more complex patterns. They're vital in various areas of mathematics and science, helping solve real-world problems.

A sequence is defined by a rule that determines the subsequent numbers based on the position in the sequence. It can be finite or infinite, with sequences such as arithmetic providing a straightforward linear pattern.

Other common sequences include:

• Geometric sequences: Each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
• Fibonacci sequences: A sequence where each number is the sum of the two preceding ones, starting from 0 and 1.

Mathematical sequences like these are foundational tools in understanding series, calculus, and even computer algorithms. They offer a way to model patterns and changes systematically.

Term calculation

In order to calculate terms of a sequence, you must understand the formula that defines the sequence. For sequences given by a specific formula, each term can be easily found by plugging in the corresponding position number.

With the example given in the original exercise, the sequence is defined by the formula:

• \( a_{n} = \frac{2n+1}{n^3} \)

This means for any term \( a_n \), simply substitute \( n \) with the position number. This allows for systematic term calculation:

• Using \( n = 1 \), calculate the first term.
• Using \( n = 2 \), calculate the second term, and so on.

Calculating terms in a sequence helps identify patterns and understand the sequence more comprehensively. It also aids in predicting further terms beyond the initial few, giving insights into the long-term behavior of the sequence.