Problem 33 - Algebra 2 and Trigonometry

Question

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=1, a_{n}=2 a_{n-1}+1 $$

Step-by-Step Solution

Answer

The first five terms are 1, 3, 7, 15, 31.

Step 1: Understanding the Sequence

We are given a recursive formula for the sequence: \(a_{1} = 1\) and \(a_{n} = 2a_{n-1} + 1\). This means each term depends on the previous term.

Step 2: Calculate the First Term

The first term, \(a_1\), is already given as 1.

Step 3: Calculate the Second Term

Use the recursive formula to find the second term. Substitute \(a_1\) into the formula: \[ a_2 = 2 \times a_1 + 1 = 2 \times 1 + 1 = 3 \]

Step 4: Calculate the Third Term

Use the recursive formula with the second term to find the third term: \[ a_3 = 2 \times a_2 + 1 = 2 \times 3 + 1 = 7 \]

Step 5: Calculate the Fourth Term

Use the recursive formula with the third term to find the fourth term: \[ a_4 = 2 \times a_3 + 1 = 2 \times 7 + 1 = 15 \]

Step 6: Calculate the Fifth Term

Use the recursive formula with the fourth term to find the fifth term: \[ a_5 = 2 \times a_4 + 1 = 2 \times 15 + 1 = 31 \]

Key Concepts

Sequence Terms

In a recursive sequence, each term is built from the previous one. This means we start with a given number, known as the first term of the sequence, and then use a specific rule to find all subsequent numbers. The term "sequence terms" refers to these individual numbers in the sequence.

For example, consider a sequence starting with a first term of 1, noted as \(a_1 = 1\). Each consecutive term \(a_n\) is calculated based on the prior term \(a_{n-1}\). In our specific problem, the recursive formula \(a_n = 2a_{n-1} + 1\) gives us each new term. By following the rule, we build the sequence term by term. This approach aids in understanding how sequences grow or change, often predicting the behavior of future terms.

Recursive Formula

The magic behind recursive sequences is the recursive formula. This formula gives us a way to find each term based on the previous one, making the sequence dynamic and often quite interesting.

The recursive formula is essential because it defines the sequence's operation — how each term links to its predecessor. Let's break down the recursive formula we have:

• \(a_1 = 1\): This is called the base case or initial term.
• \(a_n = 2a_{n-1} + 1\): This shows how every term after the first is calculated.

The beauty of a recursive formula is that it condenses seemingly complex sequences into simpler calculations. Through every recursive step, we maintain a pattern and build our sequence without needing to know all terms at once.

Mathematical Patterns

Finding and understanding patterns in mathematics is crucial, especially with sequences. Patterns help us see the behavior in numbers and predict future terms without having to calculate each one manually.

With the sequence generated by our recursive formula, we observe a simple mathematical pattern. Each term seems to double the previous term and then add 1. This specific pattern, produced by the formula \(a_n = 2a_{n-1} + 1\), simplifies predictions for the next terms once the structure is recognized.

• This doubling effect is a multiplication pattern.
• The consistent addition of 1 is an arithmetic pattern.

By identifying such patterns, students can better navigate sequences and explore connections within algebraic concepts.

Algebra 2

In an Algebra 2 context, recursive sequences serve as a powerful teaching tool for understanding complex algebraic relationships. Algebra 2 expands on foundational knowledge, integrating concepts like sequences to describe real-world problems and abstract numerical ideas.

Students in Algebra 2 learn to recognize how recursive formulas provide a routine method for constructing sequences. This topic not only covers arithmetic and geometric sequences but also delves into more complex forms, such as the one highlighted in this exercise.

• Recursive sequences bridge earlier algebra topics with higher-level mathematics.
• They illustrate how formulas are applied in iterative procedures.

By understanding the mechanics of recursive sequences, students learn important skills for solving problems, understanding patterns, and applying these ideas to varied mathematical situations.