Problem 26 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The explicit formula is \( a_m = 4 \) for all \( m \geq 1 \). Induction proves it is valid.

Step 1: Understand the Recursive Formula

The given recursive sequence is described by the relations \( a_{n+1} = 3a_n - 8 \) with the initial term \( a_1 = 4 \). This indicates each term is obtained by multiplying the previous term by 3 and then subtracting 8.

Step 2: Identify a Pattern

Calculate the first few terms to discern a pattern:- \( a_1 = 4 \)- \( a_2 = 3 \times 4 - 8 = 4 \)- \( a_3 = 3 \times 4 - 8 = 4 \)The sequence is constant with all terms equal to 4.

Step 3: Formulate the Explicit Formula

From observing the pattern, we hypothesize that the sequence is constant, leading to the explicit formula: \( a_m = 4 \) for all \( m \geq 1 \).

Step 4: Prove the Explicit Formula by Induction

Using induction to prove \( a_m = 4 \):1. **Base case**: When \( m = 1 \), \( a_1 = 4 \) which satisfies the formula.2. **Inductive step**: Assume for some \( k \geq 1 \), \( a_k = 4 \). We need to show that \( a_{k+1} = 4 \). - From the recursive formula, \( a_{k+1} = 3a_k - 8 \). - Substitute \( a_k = 4 \) into the equation: \( a_{k+1} = 3 \times 4 - 8 = 12 - 8 = 4 \). - Therefore, \( a_{k+1} = 4 \) which completes the inductive step.

Key Concepts

Explicit Formula

Finding an explicit formula involves identifying a direct relationship between the term's position in the sequence and its value. In a recursive sequence, each term is expressed as a function of previous terms. However, an explicit formula allows you to compute the nth term without needing to know the preceding terms.

In this particular exercise, the recursive definition provided is: \( a_{n+1} = 3a_n - 8 \) with an initial term \( a_1 = 4 \). By evaluating the first few terms, we observe that \( a_2 = 4 \) and \( a_3 = 4 \). This leads us to hypothesize that the sequence is constant at 4 for all terms, i.e., \( a_m = 4 \) for every \( m \geq 1 \).

The notion of pattern recognition is key when formulating an explicit formula. Here, the consistent results of 4 suggest a formula can indeed be simple due to repeated results.

• Allows direct calculation of any term.
• Doesn't depend on preceding terms.
• Often requires looking for patterns in the sequence.

Mathematical Induction

Mathematical induction is a powerful tool used for proving that a formula is valid for all terms in a sequence. It's a two-step process that ensures the formula applies not just to a few initial cases but to all possible cases in the series.

**The Base Case**: Start by proving that the formula works for the initial term of the sequence. Here, the base case is to show that \( a_1 = 4 \) satisfies \( a_m = 4 \). Indeed, \( a_1 = 4 \) which agrees with our explicit formula.

**The Inductive Step**: Assuming the formula holds for a particular term \( a_k = 4 \), you demonstrate that \( a_{k+1} = 4 \) also holds true. According to our recursive rule, substituting \( a_k = 4 \) into the recursion gives \( a_{k+1} = 3(4) - 8 = 4 \). Thus, induction confirms the consistency of the explicit formula for all terms.

This process:

• Begins with a known truth (base case).
• Uses the assumption for single cases to prove the next (inductive step).
• Ensures the formula holds universally in the sequence.

Initial Term

The initial term forms the starting point of any sequence, especially in a recursive sequence. Without this foundational term, there would be no sequence development as each term depends on the preceding term.

In this exercise, the initial term was given as \( a_1 = 4 \). This value is important since all subsequent terms rely on it according to the recursive formula: \( a_{n+1} = 3a_n - 8 \).

By substituting the initial term into the recursive formula:

• We found \( a_2 = 4 \), \( a_3 = 4 \), suggesting a pattern or trend.
• It helped to derive a consistent, constant sequence allowing the formulation of the explicit formula.

Understanding and having correct initial terms is essential because:

• It directly influences the behavior of the sequence.
• Acts as the cornerstone in proving general formulas.
• Offers a reference point for verifying overall formula validity when using mathematical induction.