Question

\(a_{n}=4 a_{n-1}+21 a_{n-2}\) for \(n \geq 2\) where \(a_{0}=3\) and \(a_{1}=7\).

Short answer

\(a_{2} = 91\), \(a_{3} = 511\), \(a_{4} = 3955\).

Step-by-step solution

Understand the Recurrence Relation

The problem provides a sequence defined by a recurrence relation \(a_{n}=4a_{n-1}+21 a_{n-2}\). This means each term is determined by the two preceding terms. You also have initial conditions: \(a_{0}=3\) and \(a_{1}=7\).

Calculate the Second Term

Using the given initial conditions, calculate the second term \(a_{2}\) using the recurrence relation:\[a_{2} = 4a_{1} + 21a_{0}\]Substitute \(a_{0} = 3\) and \(a_{1} = 7\):\[a_{2} = 4 \times 7 + 21 \times 3 = 28 + 63 = 91\]

Calculate the Third Term

Now, calculate the third term \(a_{3}\) using the recurrence relation:\[a_{3} = 4a_{2} + 21a_{1}\]Substitute \(a_{1}=7\) and \(a_{2}=91\):\[a_{3} = 4 \times 91 + 21 \times 7 = 364 + 147 = 511\]

Calculate the Fourth Term

Use the recurrence relation for \(a_{4}\):\[a_{4} = 4a_{3} + 21a_{2}\]Substitute \(a_{2} = 91\) and \(a_{3} = 511\):\[a_{4} = 4 \times 511 + 21 \times 91 = 2044 + 1911 = 3955\]

Verify and Conclude

So far, we have calculated the first few terms: \(a_{0} = 3\), \(a_{1} = 7\), \(a_{2} = 91\), \(a_{3} = 511\), and \(a_{4} = 3955\). The calculations follow the recurrence and initial conditions correctly.

Key concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematical sciences that deals with distinct and separate values or objects. It involves mathematical structures that are fundamentally discrete rather than continuous. Examples include integers, graphs, and logical statements. In discrete mathematics, elements are countable and can take only specific, separated values. This is different from continuous mathematics, where values can range smoothly.

Recurrence relations, such as the one in our exercise, are an essential part of discrete mathematics. They describe sequences in which each term is derived from one or more preceding ones.

• Elementarily, they represent how something changes over time or under some conditions, using specific rules.
• They are widely used in computer science, particularly in algorithms and data structures, to describe the time or space complexity of recursive procedures.
• They also appear in other areas like economics for modeling processes or flows that evolve step-wise over time.

Sequence Calculation

Calculating the sequence defined by a recurrence relation involves determining the terms based on the initial conditions and the recurrence formula. In the given example, the sequence is defined by the recurrence equation: \(a_{n} = 4a_{n-1} + 21a_{n-2}\).

To calculate a specific term in the sequence, you follow these steps:

• Begin with the initial terms, which are provided: here, \(a_{0} = 3\) and \(a_{1} = 7\).
• Use the recurrence relation to calculate subsequent terms. This involves substituting the known terms into the equation.
• Each term relies on the values of the previous terms, which is why starting with initial conditions is critical.

For example, to find \(a_{2}\), you use \(a_{1}\) and \(a_{0}\): \[a_{2} = 4a_{1} + 21a_{0} = 4 \times 7 + 21 \times 3 = 28 + 63 = 91\]. Sequence calculation is systematic and relies on following the established pattern or rule, ensuring each new term is correctly derived from its predecessors.

Initial Conditions

Initial conditions in recurrence relations serve as the starting point for generating the sequence of numbers. They define specific values for the sequence at the beginning, allowing us to compute all subsequent terms.

In our exercise, the initial conditions are \(a_{0} = 3\) and \(a_{1} = 7\). These are crucial because:

• They anchor the start of the sequence.
• Without these values, the rest of the terms could not be calculated as the formula depends on previous terms.
• This concept also ensures the uniqueness of the sequence: different initial conditions lead to entirely different sequences, even if the recurrence formula remains the same.

Initial conditions not only set the sequence but also can be interpreted as the foundation of a recursive approach, often corresponding to base cases in recursive algorithms in computer science.