Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

\(a_{1}=2, \quad a_{k+1}=3 a_{k}-5\)

Step-by-Step Solution

Answer

Calculate the sequence terms using the recursion formula.

Step 1: Understanding the Problem

We are given a recursive sequence where the initial term is \( a_1 = 2 \) and each subsequent term is defined by \( a_{k+1} = 3a_k - 5 \). Our task is to find the terms of the sequence using this recursive formula.

Step 2: Calculating the Second Term

To find \( a_2 \), we use \( a_1 \) in the formula: \( a_2 = 3a_1 - 5 \). Substituting \( a_1 = 2 \), we get \( a_2 = 3(2) - 5 = 6 - 5 = 1 \).

Step 3: Calculating the Third Term

Using the previously calculated term \( a_2 \), we find \( a_3 \) using the formula: \( a_3 = 3a_2 - 5 \). Substitute \( a_2 = 1 \): \( a_3 = 3(1) - 5 = 3 - 5 = -2 \).

Step 4: Calculating the Fourth Term

Next, we use \( a_3 \) to find \( a_4 \): \( a_4 = 3a_3 - 5 \). Substitute \( a_3 = -2 \): \( a_4 = 3(-2) - 5 = -6 - 5 = -11 \).

Step 5: Calculating the Fifth Term

Finally, using \( a_4 \), calculate \( a_5 \): \( a_5 = 3a_4 - 5 \). Substitute \( a_4 = -11 \): \( a_5 = 3(-11) - 5 = -33 - 5 = -38 \).

Key Concepts

Recursive Formula

In mathematics, a recursive formula is a powerful tool used to define terms in a sequence based on previous terms. For a sequence defined by a recursive formula, to find any term, you apply the operation or rule to the preceding term. In our exercise, the recursive formula is given as \(a_{k+1} = 3a_k - 5\). This tells us how any term \(a_{k+1}\) can be determined by its preceding term \(a_k\).

One key thing to understand about recursive formulas is the importance of the initial condition, often known as the "initial term." In our problem, the initial term is \(a_1 = 2\). This initial term serves as the starting point from which all subsequent terms are generated. Without this starting point, the recursive process would not begin. Recursive formulas generate sequences by gradually computing values, always building upon the previous term calculated. This method is very effective for sequences where a clear pattern emerges as you proceed.

Sequence Terms

Sequence terms refer to the individual elements or figures that make up a sequence. In this context, each number in the sequence is calculated using the recursive formula. Starting from the initial term, which is \(a_1 = 2\), we use the recursive formula to find each subsequent term.

For our sequence, the computations are as follows:

• The first term, \(a_1\), is 2.
• The second term, \(a_2\), is calculated using the formula:\( a_2 = 3a_1 - 5 = 1\).
• The third term, \(a_3\), follows by the formula, yielding: \(a_3 = 3a_2 - 5 = -2\).
• The fourth term, \(a_4\), is obtained from: \(a_4 = 3a_3 - 5 = -11\).
• The fifth term, \(a_5\), results in: \(a_5 = 3a_4 - 5 = -38\).

Sequence terms often show interesting behaviors or patterns, especially when examined over a number of iterations. Whether they increase, decrease, or oscillate can give insights into the underlying characteristics of the recursive rule at play. Understanding these terms individually and their relations helps to see the flow of the sequence.

Mathematical Induction

Mathematical induction is a method of mathematical proof used extensively in sequences and other areas of discrete mathematics. It helps in proving that a statement holds true for all natural numbers. Though not directly used in calculating each term in our sequence exercise, mathematical induction could be applied to prove properties about the sequence in general.

To apply mathematical induction, you start with two key steps:

• Base Case: Verify that a statement holds for the first term of the sequence. Often, this involves substituting the initial term and checking if the condition satisfies.
• Inductive Step: Assume the statement is true for an arbitrary term \(a_k\). Then demonstrate it must also be true for the next term \(a_{k+1}\).

For instance, if we wanted to prove that all terms \(a_n\) in our sequence are odd, we would first show that the initial term \(a_1=2\) is even (which wouldn't support the odd statement, thus can't use). If the hypothesis were different and the criteria for being even were needed, you show \(a_k\) implies \(a_{k+1}\) holds true within the recursive framework. Understanding induction deepens insight into the behavior of a sequence and supports validating broader claims about recursive sequences.