Question

Find the first six terms of the sequence with the elements defined as \(F(0)=1, F(1)=\) 3, \(F(2)=5,\) and \(F(n)=3 F(n-1)+2 F(n-2)-3 F(n-3)\) for \(n \geq 3 .\)

Short answer

The first six terms are 1, 3, 5, 18, 55, 186, and 614.

Step-by-step solution

Understanding Initial Terms

We start by identifying the initial terms of the sequence. Given are: \( F(0) = 1 \), \( F(1) = 3 \),\( F(2) = 5 \). These will help in calculating further terms based on the recursive formula provided.

Recursive Formula Application

Apply the recursive formula \( F(n) = 3F(n-1) + 2F(n-2) - 3F(n-3) \) to find the subsequent terms starting from \( n = 3 \).

Calculate \( F(3) \)

Substitute \( F(0), F(1) \) and \( F(2) \) into the formula:\[ F(3) = 3 \cdot F(2) + 2 \cdot F(1) - 3 \cdot F(0) \]\[ = 3 \cdot 5 + 2 \cdot 3 - 3 \cdot 1 \]\[ = 15 + 6 - 3 = 18 \]

Calculate \( F(4) \)

Use \( F(1), F(2) \) and \( F(3) \) to find \( F(4) \):\[ F(4) = 3 \cdot F(3) + 2 \cdot F(2) - 3 \cdot F(1) \]\[ = 3 \cdot 18 + 2 \cdot 5 - 3 \cdot 3 \]\[ = 54 + 10 - 9 = 55 \]

Calculate \( F(5) \)

Calculate \( F(5) \) using \( F(2), F(3) \) and \( F(4) \):\[ F(5) = 3 \cdot F(4) + 2 \cdot F(3) - 3 \cdot F(2) \]\[ = 3 \cdot 55 + 2 \cdot 18 - 3 \cdot 5 \]\[ = 165 + 36 - 15 = 186 \]

Calculate \( F(6) \)

Finally, calculate \( F(6) \) using \( F(3), F(4) \) and \( F(5) \):\[ F(6) = 3 \cdot F(5) + 2 \cdot F(4) - 3 \cdot F(3) \]\[ = 3 \cdot 186 + 2 \cdot 55 - 3 \cdot 18 \]\[ = 558 + 110 - 54 = 614 \]

Conclusion

The first six terms of the sequence are: \( F(0) = 1, F(1) = 3, F(2) = 5, F(3) = 18, F(4) = 55, F(5) = 186, F(6) = 614 \).

Key concepts

Sequence Calculation

Calculating the terms in a sequence, especially when it's defined recursively, involves understanding how each term is built upon the previous ones. The objective is to find the first few terms using a given starting point (initial conditions) and a specific rule or formula (recursive formula).

In the given exercise, the sequence starts with three initial terms: \( F(0) = 1 \), \( F(1) = 3 \), and \( F(2) = 5 \). These are important as they anchor the sequence, serving as the basis for calculating subsequent terms. Each new term from \( n = 3 \) onwards is calculated using the formula \( F(n) = 3F(n-1) + 2F(n-2) - 3F(n-3) \).

Understanding this process means recognizing how each term becomes a building block for the next. Sequentially plugging the values into the recursive formula allows us to uncover each term step by step.

Recursive Formula

A recursive formula, like the one given in the exercise, provides a way to calculate any term in a sequence based on a number of preceding terms. It's a bit like a recipe where you need prior steps to advance to the next.

The exercise uses the recursive formula \( F(n) = 3F(n-1) + 2F(n-2) - 3F(n-3) \). This means each term depends on the three terms that came before it.

• 3F(n-1) uses the term directly before the target term, multiplied by 3.
• 2F(n-2) involves the term two places back, multiplied by 2.
• -3F(n-3) takes the term three positions back, multiplied by -3, which reduces the influence of older terms.

By recursively applying this formula with the initial terms, you can compute each subsequent term in the sequence.

Initial Conditions

Initial conditions in a sequence are the starting values that allow the recursive process to begin effectively. Without these, we cannot calculate further terms using a recursive formula.

In our example, the initial conditions are given as \( F(0) = 1 \), \( F(1) = 3 \), and \( F(2) = 5 \). These terms initialize the sequence and provide the essential data points needed for the recursive formula to generate additional terms.

• They set the scene: ensure our sequence can start and proceed correctly.
• They are crucial for correct computation: any error in these values will propagate throughout the sequence.

Understanding the importance of initial conditions helps in appreciating how sequences develop and why every term relies on both the formula and these starting terms to build the sequence correctly. They are, in essence, the foundation of the calculation process.