Problem 18 - Algebra and Trigonometry

Question

List the first five terms of each sequence. \(\left\\{b_{n}\right\\}=\left\\{\frac{2 n+1}{2 n}\right\\}\)

Step-by-Step Solution

Answer

The first five terms are \( \frac{3}{2} \), \( \frac{5}{4} \), \( \frac{7}{6} \), \( \frac{9}{8} \), and \( \frac{11}{10} \).

Step 1 - Understand the Sequence

The given sequence is defined by the formula \( b_n = \frac{2n + 1}{2n} \). This formula will be used to calculate the terms by substituting different values of \( n \).

Step 2 - Calculate for n=1

Substitute \( n = 1 \) into the formula: \[ b_1 = \frac{2(1) + 1}{2(1)} = \frac{3}{2}. \]

Step 3 - Calculate for n=2

Substitute \( n = 2 \) into the formula: \[ b_2 = \frac{2(2) + 1}{2(2)} = \frac{5}{4}. \]

Step 4 - Calculate for n=3

Substitute \( n = 3 \) into the formula: \[ b_3 = \frac{2(3) + 1}{2(3)} = \frac{7}{6}. \]

Step 5 - Calculate for n=4

Substitute \( n = 4 \) into the formula: \[ b_4 = \frac{2(4) + 1}{2(4)} = \frac{9}{8}. \]

Step 6 - Calculate for n=5

Substitute \( n = 5 \) into the formula: \[ b_5 = \frac{2(5) + 1}{2(5)} = \frac{11}{10}. \]

Key Concepts

arithmetic sequences

An arithmetic sequence is a series of numbers in which each term is derived by adding a fixed amount, called the common difference, to the previous term. For example, the sequence 2, 4, 6, 8, 10 is an arithmetic sequence with a common difference of 2. Understanding these sequences is important because they often appear in various math problems.
Although the given sequence \( b_n = \frac{2n+1}{2n} \) is not an arithmetic sequence, knowing how arithmetic sequences work can help you identify other types of sequences. They share the idea of having a recognizable pattern where one term depends on the previous one.

substitution method

The substitution method involves replacing a variable with a specific value to simplify and solve an equation. This method is especially useful in sequences where a general formula is given. Each term is found by plugging in integer values successively.
For example, in the given sequence \( b_n = \frac{2n + 1}{2n} \), you calculate the first term by substituting \( n = 1 \): \[ b_1 = \frac{2(1) + 1}{2(1)} = \frac{3}{2}. \]
Repeat this process for \( n = 2 \) to get \[ b_2 = \frac{2(2) + 1}{2(2)} = \frac{5}{4}. \] Continue this pattern to find additional terms of the sequence. It's a straightforward approach that makes complex expressions simpler and more manageable.

fractional expressions

Fractional expressions involve numerators and denominators, which are expressions themselves. Understanding these is key because they frequently appear in sequences and various other algebraic contexts. In the sequence \( b_n = \frac{2n + 1}{2n} \), each term is a fraction.
Let's break down the calculation process:

• Substitute \( n \) with 1: \[ b_1 = \frac{2(1) + 1}{2(1)} = \frac{3}{2}. \]
• Substitute \( n \) with 2: \[ b_2 = \frac{2(2) + 1}{2(2)} = \frac{5}{4}. \]
• Substitute \( n \) with 3: \[ b_3 = \frac{2(3) + 1}{2(3)} = \frac{7}{6}. \]
• Substitute \( n \) with 4: \[ b_4 = \frac{2(4) + 1}{2(4)} = \frac{9}{8}. \]
• Substitute \( n \) with 5: \[ b_5 = \frac{2(5) + 1}{2(5)} = \frac{11}{10}. \]

The numerators and denominators can be algebraic expressions, which you simplify by substituting values step-by-step. Knowing how to handle fractions will help you better understand more advanced algebra concepts.