Problem 26 - Precalculus: Mathematics for Calculus

Question

A sequence is defined recursively by \(a_{n+1}=3 a_{n}-8\) and \(a_{1}=4 .\) Find an explicit formula for \(a_{n},\) and then use mathematical induction to prove that the formula you found is true.

Step-by-Step Solution

Answer

The sequence is constant: \( a_n = 4 \) for all \( n \geq 1 \).

Step 1: Identify the Given Recurrence Relation

We are given the recursive relation \( a_{n+1} = 3a_n - 8 \) and the initial condition \( a_1 = 4 \). Our task is to find an explicit formula for \( a_n \).

Step 2: Formulate Hypothesis for Explicit Formula

Assume that the explicit formula is in the form \( a_n = A\cdot B^n + C \). We need to determine the values of \( A \), \( B \), and \( C \).

Step 3: Substitute into Recurrence Relation

Substitute \( a_{n+1} = 3a_n - 8 \) and \( a_n = A\cdot B^n + C \) into the recurrence relation, resulting in \( A\cdot B^{n+1} + C = 3(A \cdot B^n + C) - 8 \).

Step 4: Equating Terms

Equate the coefficients for \( B^{n+1} \) and constant terms: we get \( AB = 3A \) and \( C = 3C - 8 \). Solve these equations to find \( B = 3 \) and using the equation for C: \( 2C = 8 \), yielding \( C = 4 \).

Step 5: Verify Initial Condition and Solve for A

Substitute \( a_1 = 4 \) into the explicit formula \( A \cdot 3^1 + 4 = 4 \). Calculate \( A \times 3 + 4 = 4 \) which gives \( A = 0 \).

Step 6: Obtain the Explicit Formula

The explicit formula for the sequence becomes \( a_n = 4 \cdot 0^n + 4 \), which simplifies to \( a_n = 4 \). This suggests that the sequence is constant.

Step 7: Prove by Mathematical Induction

**Base Case:** For \( n=1 \), \( a_1=4 \), which matches the initial condition. **Inductive Step:** Assume \( a_k = 4 \) holds. Substitute \( a_{k+1} = 3 \cdot 4 - 8 \) which simplifies to \( a_{k+1} = 4 \). By the principle of induction, the explicit formula \( a_n = 4 \) holds for all \( n \geq 1 \).

Key Concepts

Recursive Sequences

Recursive sequences are a particular type of sequence where each term is defined based on its preceding terms. That means to find any term in the sequence, you need to know one or more of the previous terms. These sequences often come with initial conditions that help standardize how the sequence begins.

In our given exercise, the sequence is described by a recurrence relation:

• The formula for the sequence is \( a_{n+1} = 3a_n - 8 \). This tells us that to find the next term \( a_{n+1} \), we multiply the current term \( a_n \) by 3 and then subtract 8.
• We also have the initial term \( a_1 = 4 \). This starting point is crucial because it allows us to generate further terms using the recurrence relation.

The power of recursive sequences lies in their ability to model situations where each outcome depends on the previous one, like in investment growth over time or population models. However, sometimes finding an explicit formula for them can simplify understanding of the sequence's behavior.

Explicit Formulas

An explicit formula provides a direct way to determine any term in a sequence without having to know the previous term. This makes it much easier to compute terms of the sequence, especially for large indices.

In the given problem, we were tasked to find an explicit formula for our recursive sequence. Starting with the hypothesis that it has the form \( a_n = A \cdot B^n + C \), we utilized the given recursive relation to determine that:

• \( B = 3 \) because multiplying any term by \( B \) should ideally produce the next term before adjustments with \( C \).
• \( C = 4 \) as determined by satisfying the equation \( C = 3C - 8 \). Solving gives \( C = 4 \).

Finally, using the initial condition \( a_1 = 4 \), we determined \( A = 0 \). This simplified the explicit formula to \( a_n = 4 \), indicating the sequence is constant at 4. An explicit formula like this is highly valuable as it provides immediate insight into the sequence's overall behavior.

Mathematical Induction

Mathematical induction is a powerful technique used to prove statements that are formulated with a specific pattern or rule, like our explicit formula for sequences.

In the context of the exercise, we used mathematical induction to prove the correctness of our found explicit formula \( a_n = 4 \):

• **Base Case:** We first established the base case for \( n=1 \), where \( a_1 = 4 \). This step ensures that the formula is true for the initial term of the sequence.
• **Inductive Step:** We then assumed the formula is true for some arbitrary term \( a_k \), i.e., \( a_k = 4 \). With this assumption, we showed that it follows that \( a_{k+1} = 4 \) when using our recursive relation.

By the induction principle, since both the base case and the inductive step were satisfied, the formula \( a_n = 4 \) is proven to be valid for all integer values of \( n \). Inductive proofs are essentially about chain reactions - setting off a domino effect by confirming the next term follows as expected from the formula.