Problem 13 - Algebra and Trigonometry

Question

Find the first five terms of the given recursively defined sequence. \(a_{n}=2 a_{n-1}+1 \quad\) and \(\quad a_{1}=1\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Recursive Formula

The problem gives us a recursive formula: \( a_{n} = 2a_{n-1} + 1 \). This means that to find any term \( a_n \), we need to know the value of the previous term \( a_{n-1} \). Additionally, we are given the initial condition \( a_1 = 1 \). Our task is to apply this formula iteratively to find the sequence's terms.

Step 2: Compute the Second Term

Start with the initial term \( a_1 = 1 \). Use the recursive formula to find \( a_2 \): \( a_2 = 2a_1 + 1 = 2(1) + 1 = 3 \). So, the second term is \( a_2 = 3 \).

Step 3: Compute the Third Term

Using \( a_2 = 3 \), apply the recursive formula to find \( a_3 \): \( a_3 = 2a_2 + 1 = 2(3) + 1 = 7 \). Therefore, the third term is \( a_3 = 7 \).

Step 4: Compute the Fourth Term

Now that we know \( a_3 = 7 \), use this to find \( a_4 \): \( a_4 = 2a_3 + 1 = 2(7) + 1 = 15 \). So, the fourth term is \( a_4 = 15 \).

Step 5: Compute the Fifth Term

With \( a_4 = 15 \), apply the recursive relationship to determine \( a_5 \): \( a_5 = 2a_4 + 1 = 2(15) + 1 = 31 \). Thus, the fifth term is \( a_5 = 31 \).

Key Concepts

Recursion

Recursion in mathematics is a process of defining a sequence or function where each term or value is derived from the preceding ones. It involves a base case and a recursive formula. In the problem we are tackling, this concept is front and center. The equation given is recursive:

• Base case: First term is defined without depending on other terms, here it's given that \( a_1 = 1 \).
• Recursive formula: Each subsequent term is calculated using the previous term, given by \( a_n = 2a_{n-1} + 1 \).

The beauty of recursion lies in its simplicity and elegance. By defining how a sequence builds upon itself, a potentially infinite set of numbers can be predictably generated.

Sequence Terms

Sequence terms refer to the individual elements or entries within a sequence, which can be thought of as a list of numbers or objects. In our exercise, terms are generated from a starting value using a fixed rule.
Each term in a sequence can be calculated using the formula given. Let's break them down to understand:

• Initial Term: The sequence starts with \( a_1 = 1 \), which is independently defined.
• Subsequent Terms: Each new term \( a_n \) is found using the preceding term \( a_{n-1} \) with the formula \( a_n = 2a_{n-1} + 1 \).

This iterative generation of terms is central to understanding and applying recursive sequences. Comprehending each term's relation to its predecessor reinforces the sequence's growth pattern.

Step-by-Step Solution

Approaching a recursively defined sequence calls for a clear and organized method. Let's go through the process of determining the first five sequence terms using a systematic sequence of steps:

• Step 1: Start with what's given, \( a_1 = 1 \).
• Step 2: Compute \( a_2 \) using the formula, \( a_2 = 2a_1 + 1 = 3 \).
• Step 3: Use \( a_2 \) to find \( a_3 \), \( a_3 = 2a_2 + 1 = 7 \).
• Step 4: With \( a_3 = 7 \), determine \( a_4 \), \( a_4 = 2a_3 + 1 = 15 \).
• Step 5: Apply the relation for \( a_5 \), \( a_5 = 2a_4 + 1 = 31 \).

By systematically following these steps, one can unravel the complexity of the sequence, ensuring each term is logically derived from its predecessor. This methodical approach not only finds the correct terms but also solidifies understanding of the recursive sequence itself.