Problem 82 - Algebra for College Students

Question

$$ \left\\{\begin{array}{l} a_{1}=2 \\ a_{2}=3 \\ a_{n}=2 a_{n-2}+3 a_{n-1} \quad \text { for } n \geq 3 \end{array} \quad 2,3,13,45,161,573\right. $$

Step-by-Step Solution

Answer

The sequence satisfies the recurrence relation and given initial values.

Step 1: Understand the Sequence Rules

The given sequence is defined by an initial two terms: \( a_1 = 2 \) and \( a_2 = 3 \), and a recurrence relation \( a_n = 2a_{n-2} + 3a_{n-1} \) for \( n \geq 3 \). This means to find any term we need the two preceding terms.

Step 2: Validate the Given Sequence

Using the recurrence relation, validate the terms:- For \( n=3 \): \( a_3 = 2a_1 + 3a_2 = 2 \times 2 + 3 \times 3 = 4 + 9 = 13 \).- For \( n=4 \): \( a_4 = 2a_2 + 3a_3 = 2 \times 3 + 3 \times 13 = 6 + 39 = 45 \).- For \( n=5 \): \( a_5 = 2a_3 + 3a_4 = 2 \times 13 + 3 \times 45 = 26 + 135 = 161 \).- For \( n=6 \): \( a_6 = 2a_4 + 3a_5 = 2 \times 45 + 3 \times 161 = 90 + 483 = 573 \).The provided terms match the sequence generated by the rule.

Step 3: Confirm Completion

Each term from \( a_1 \) to \( a_6 \) has been validated using the recurrence relation, confirming that the sequence 2, 3, 13, 45, 161, 573 follows the defined rules without error.

Key Concepts

Sequence Validation

Validating a sequence involves checking if the terms of a sequence follow the given rule or equation. In the context of recursive sequences, this often means using the recurrence relation to check whether each term fits.
To validate a sequence like the one given here, follow these steps:

• Start with the initial terms provided.
• Apply the recurrence relation to these terms to compute subsequent terms.
• Compare these computed terms with the ones given in the sequence.

For example, in our sequence, we began with the terms 2 and 3. The next terms are calculated using the formula:\( a_n = 2a_{n-2} + 3a_{n-1} \)
By calculating each term step by step and checking against the initial sequence, we validated that the sequence 2, 3, 13, 45, 161, 573 indeed follows the defined rules. Each calculation aligns, confirming sequence validity. It’s a trustworthy way to ensure no errors exist in the sequence definition.

Initial Terms

Initial terms in a sequence are the starting numbers used to begin a recursive process. They are essential because, without them, it is impossible to proceed with any calculations that follow the recurrence relation.
When given initial terms like the ones in this sequence, 2 and 3, they set the foundation for deriving all future terms. They serve as a starting point, and every subsequent term depends on them.

• Foundation: Initial terms are the sequence's backbone, needed to compute every other term.
• Uniqueness: Different initial terms will result in different sequences, even with the same recurrence relation.

In our case, we began with these initial terms and used them to derive the entire sequence through the recurring formula \( a_n = 2a_{n-2} + 3a_{n-1} \). Without the precision of these first values, the entire sequence could be miscalculated.

Recursive Sequences

Recursive sequences are mathematical sequences where each term is a function of one or more previous terms. This dependency on prior terms is defined through a recurrence relation.
In our exercise, the sequence is defined recursively using the rule:\( a_n = 2a_{n-2} + 3a_{n-1} \)
where each new term is computed from two earlier terms in the sequence.

• Predictability: The sequence's future is predictable as long as the initial terms and the recurrence relation are consistent.
• Complexity: Recursive sequences can quickly become complex, but understanding their dependence on initial terms can simplify this complexity.

They are powerful tools in various mathematical and practical applications, such as modeling population growth or financial models. They rely on maintaining accuracy across each sequential calculation to ensure the sequence remains valid and useful.