Problem 83 - Algebra for College Students

Question

\left\\{\begin{array}{l} a_{1}=3 \\ a_{2}=1 \\ a_{n}=\left(a_{n-1}-a_{n-2}\right)^{2} \quad \text { for } n \geq 3 \end{array} \quad 3,1,4,9,25,256\right.

Step-by-Step Solution

Answer

The terms of the sequence are 3, 1, 4, 9, 25, 256.

Step 1: Understanding the Sequence

This sequence is defined recursively with two initial terms: \(a_1 = 3\) and \(a_2 = 1\). For \(n \geq 3\), subsequent terms are defined using the formula \(a_n = (a_{n-1} - a_{n-2})^2\). Let's find the first few terms to confirm the pattern.

Step 2: Calculate Third Term

We need to find the third term \(a_3\) using the formula. Substitute the values of \(a_1\) and \(a_2\):\[a_3 = (a_2 - a_1)^2 = (1 - 3)^2 = (-2)^2 = 4\]Therefore, \(a_3 = 4\).

Step 3: Calculate Fourth Term

Now calculate the fourth term using \(a_3\) and \(a_2\):\[a_4 = (a_3 - a_2)^2 = (4 - 1)^2 = 3^2 = 9\]So, \(a_4 = 9\).

Step 4: Calculate Fifth Term

To find the fifth term, use \(a_4\) and \(a_3\):\[a_5 = (a_4 - a_3)^2 = (9 - 4)^2 = 5^2 = 25\]Thus, \(a_5 = 25\).

Step 5: Calculate Sixth Term

For the sixth term, we use \(a_5\) and \(a_4\):\[a_6 = (a_5 - a_4)^2 = (25 - 9)^2 = 16^2 = 256\]Therefore, \(a_6 = 256\).

Step 6: Sequence Confirmation

After calculating the first six terms, we have: 3, 1, 4, 9, 25, 256. These calculated terms match the given terms, confirming the sequence is correctly established.

Key Concepts

Initial Conditions

Initial conditions in a recursive sequence are the starting points that allow us to begin calculating further terms in the sequence. In this example, we have two initial conditions:

• \( a_1 = 3 \)
• \( a_2 = 1 \)

These two values are crucial because they set the stage for all the subsequent terms. Without initial conditions, we wouldn't be able to apply the recurrence relation to find the next terms. They act as anchors or seeds from which all other terms within the sequence are generated. Understanding initial conditions is essential in grasping how a recursive pattern evolves. By having clear starting points, we ensure the correct application of mathematical formulas and consistent results.

Recurrence Relations

Recurrence relations define how we calculate each term in a recursive sequence based on preceding terms. In this sequence, the recurrence relation is given by: \[ a_n = (a_{n-1} - a_{n-2})^2 \] for any term where \( n \geq 3 \). This means every term from the third onward is determined by the square of the difference of the two previous terms.
The significance of recurrence relations lies in their ability to provide a formulaic method for generating a sequence. This, in turn, helps to identify patterns and predict future terms without directly computing each one. Understanding these relations is pivotal to mastering recursive sequences, as they form the backbone of the sequence's progression.

Term Calculation

Calculating terms in a recursive sequence involves applying the recurrence relations repeatedly. Let's break down the process for the first few terms beyond the initial conditions.

• **Third Term**: Calculate using the initial two terms \( a_1 = 3 \) and \( a_2 = 1 \).\[ a_3 = (a_2 - a_1)^2 = (1 - 3)^2 = 4 \]


• **Fourth Term**: Use \( a_3 = 4 \) and \( a_2 = 1 \).\[ a_4 = (a_3 - a_2)^2 = (4 - 1)^2 = 9 \]


• **Fifth Term**: Now use \( a_4 = 9 \) and \( a_3 = 4 \).\[ a_5 = (a_4 - a_3)^2 = (9 - 4)^2 = 25 \]


• **Sixth Term**: Continue with \( a_5 = 25 \) and \( a_4 = 9 \).\[ a_6 = (a_5 - a_4)^2 = (25 - 9)^2 = 256 \]

By understanding how to apply the recurrence relation, you can systematically find any term in the sequence. It's a step-by-step process that unravels the power of patterns in mathematics.

Mathematical Patterns

Recognizing mathematical patterns in sequences is a powerful tool that helps students predict future terms and understand the behavior of sequences. In the given recursive sequence, the pattern emerges from both the initial conditions and the recurrence relation:

• Starting with the initial conditions \( a_1 = 3 \) and \( a_2 = 1 \), the sequence unfolds into: 3, 1, 4, 9, 25, 256.
• Observe that each term grows significantly, showcasing exponential growth, as calculated by taking the difference of two previous terms and squaring it.

Mathematical patterns provide insight into how the sequence evolves over time. Recognizing such patterns not only aids in solving homework problems but also enhances the ability to approach complex mathematical challenges creatively. By studying these patterns, students can develop a more intuitive understanding of numbers and their relationships.