Problem 15 - Precalculus Mathematics for Calculus

Question

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: Understand the given sequence

The sequence is recursively defined with the formula \(a_{n} = 2a_{n-1} + 1\), meaning each term depends on the previous term. We are also given \(a_{1} = 1\) as our starting point.

Step 2: Calculate the second term \(a_2\)

Using the recursive formula \(a_{n} = 2a_{n-1} + 1\), we can find \(a_2\). Substitute \(a_1 = 1\): \(a_2 = 2 \times 1 + 1 = 3\).

Step 3: Calculate the third term \(a_3\)

Use the recursive formula again, substituting \(a_2 = 3\): \(a_3 = 2 \times 3 + 1 = 7\).

Step 4: Calculate the fourth term \(a_4\)

Continuing with the recursive formula, substitute \(a_3 = 7\): \(a_4 = 2 \times 7 + 1 = 15\).

Step 5: Calculate the fifth term \(a_5\)

Again using the recursive formula, substitute \(a_4 = 15\): \(a_5 = 2 \times 15 + 1 = 31\).

Step 6: Summarize the first five 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

Understanding Recursive Formulas

Recursive formulas are a way of defining a sequence where each term is generated based on the preceding term(s). This is different from an explicit formula, which directly defines any term in the sequence without requiring prior terms. Recursive formulas are usually beneficial in situations where each step logically builds from the last one. For example, in the given sequence, the formula is \(a_{n} = 2a_{n-1} + 1\). Here, \(a_{n-1}\) represents the previous term, and each new term \(a_{n}\) is found by plugging the preceding term into the formula and performing the arithmetic operation specified.
Therefore, to determine any term in a recursively defined sequence, we must know the initial term(s), and then continuously apply the recursive rule on each preceding term to find the subsequent ones. This is why understanding the mechanism behind the recursive formulas is key.
The magic of recursion is that it allows us to describe complex sequences with simple steps, making it efficient to work through, if the initial values are known, like in this sequence where \(a_1=1\).

Sequence Calculation

Calculating a sequence term in a recursive manner involves using the recursive formula repeatedly, starting from an initial term. In our exercise, we were given the initial term \(a_1 = 1\) and the recursive formula \(a_{n} = 2a_{n-1} + 1\). To calculate the first few terms, we followed these steps:

• Start with the initial value: \(a_1 = 1\).
• Calculate \(a_2\) by substituting \(a_1\): \(a_2 = 2\times1 + 1 = 3\).
• Use \(a_2\) to find \(a_3\): \(a_3 = 2\times3 + 1 = 7\).
• Then, use \(a_3\) for \(a_4\): \(a_4 = 2\times7 + 1 = 15\).
• Finally, calculate \(a_5\) using \(a_4\): \(a_5 = 2\times15 + 1 = 31\).

By proceeding in this step-by-step fashion, we unravel each term systematically, ensuring we account for each transition that happens at every step between successive terms.

Applying Mathematical Induction

Mathematical induction is a logical method used to prove statements or formulas that are true for all integers of a specific sequence. It works in two steps: the base case and the inductive step. Although not explicitly used in calculating our sequence, induction could verify any pattern we suspect exists in recursively defined sequences.
Here's a simplified approach:

• **Base Case:** Verify the statement for the initial value (for instance, check it for \(n=1\)). In our sequence, \(a_1 = 1\) is our base case.
• **Inductive Step:** Assume the statement holds for some arbitrary integer, say \(k\). Then, prove it holds for \(k+1\). For our sequence, assuming \(a_k = 2^k - 1\), show \(a_{k+1} = 2^{k+1} - 1\).

Induction is the bridge from verified cases to potentially infinite cases, serving as proof that a given property or pattern extends through all terms in an infinite sequence. In practice, if we suspect a formula describes our recursive sequence, induction helps substantiate that hypothesis beyond manually calculated terms.