Problem 15 - Algebra and Trigonometry

Question

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=2 a_{n-1}+1 \quad \text { and } \quad a_{1}=1 $$

Step-by-Step Solution

Answer

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

Step 1: Determine the First Term

The first term of the sequence is already provided as \(a_1 = 1\). This will serve as the base for calculating the next terms in the sequence.

Step 2: Calculate the Second Term

Use the recursive formula \(a_n = 2a_{n-1} + 1\) to find \(a_2\). Substitute \(a_1 = 1\) into the formula: \(a_2 = 2 \cdot 1 + 1 = 3\).

Step 3: Calculate the Third Term

Use \(a_2 = 3\) to find the next term. Substitute \(a_2\) into the formula: \(a_3 = 2 \cdot 3 + 1 = 7\).

Step 4: Calculate the Fourth Term

Use \(a_3 = 7\) to find \(a_4\). Substitute \(a_3\) into the formula: \(a_4 = 2 \cdot 7 + 1 = 15\).

Step 5: Calculate the Fifth Term

Use \(a_4 = 15\) to find \(a_5\). Substitute \(a_4\) into the formula: \(a_5 = 2 \cdot 15 + 1 = 31\).

Step 6: Summarize the Terms

The first five terms of the sequence are \(a_1 = 1\), \(a_2 = 3\), \(a_3 = 7\), \(a_4 = 15\), and \(a_5 = 31\).

Key Concepts

First Five Terms

When dealing with a recursively defined sequence, the goal often involves finding the first few terms to establish a pattern or understanding. A recursive definition gives us a rule that relates each term to the previous one, but it needs an initial term to get started. In our given problem, the sequence is defined by the initial term, known as the base case:

• The first term, \(a_1 = 1\).

This first term is crucial as it allows us to compute the successive terms using the recursive rule. Without it, the sequence cannot be calculated. From this starting point, we use the recursive formula to determine the subsequent terms. In our example, the first five terms have been calculated as follows:

• \(a_2 = 3\)
• \(a_3 = 7\)
• \(a_4 = 15\)
• \(a_5 = 31\)

Each of these terms builds upon the previous, thanks to the recursive definition.

Recursive Formula

The recursive formula is a crucial tool when dealing with sequences. It provides a method to find each term in a sequence by applying a specific rule to the previous term. In our current problem, the recursive formula is given by:

• \(a_n = 2a_{n-1} + 1\)

This formula tells us exactly how to find any term \(a_n\), as long as we have the value of the previous term \(a_{n-1}\). The formula consists of two major components:

• The function of the previous term—which in this case is multiplying the previous term by 2.
• A constant addition—which here is adding 1 to the result.

The beauty of recursive sequences lies in their self-dependent nature; each new term is directly connected to the one before it. This structure allows for the sequence to be easily extended as long as the previous term is known.

Sequence Calculation

Calculating the terms of a recursively defined sequence involves a systematic application of the recursive formula. Let's break down how this is done with our problem:Initially, we start with the given base case

• \(a_1 = 1\)

From here, each subsequent term is calculated using the recursive formula. Here’s how you can do it:

• To find \(a_2\), apply the formula to \(a_1\): \(a_2 = 2 imes 1 + 1 = 3\).
• To find \(a_3\), use \(a_2\): \(a_3 = 2 imes 3 + 1 = 7\).
• Keep repeating this process. For \(a_4\), use \(a_3\): \(a_4 = 2 imes 7 + 1 = 15\).
• Finally, for \(a_5\), use \(a_4\): \(a_5 = 2 imes 15 + 1 = 31\).

This sequence calculation method is what enables us to unravel the pattern locked within the recursive formula. By maintaining consistency and precision with each calculation, the terms of the sequence gradually unfold.