Problem 31 - Discrete Mathematics with Applications

Question

Using the recursive definition of \(c_{n},\) compute each. $$c_{8}$$

Step-by-Step Solution

Answer

The value of \(c_8\) using the given recursive definition is 408.

Step 1: Calculate \(c_2\)

Let's start calculating the value for \(c_2\): \(c_2 = 2c_{1} + c_0\) Since \(c_0 = 0\) and \(c_1 = 1\), we have: \(c_2 = 2 \cdot 1 + 0 = 2\)

Step 2: Calculate \(c_3\)

Next, let's calculate the value for \(c_3\): \(c_3 = 2c_{2} + c_1\) Using the values we have found earlier, we get: \(c_3 = 2 \cdot 2 + 1 = 5\)

Step 3: Keep calculating until \(c_8\)

Now that we have the values of \(c_2\) and \(c_3\), we will keep applying the recursive formula until we reach \(c_8\). \(c_4 = 2c_{3} + c_2 = 2 \cdot 5 + 2 = 12\) \(c_5 = 2c_{4} + c_3 = 2 \cdot 12 + 5 = 29\) \(c_6 = 2c_{5} + c_4 = 2 \cdot 29 + 12 = 70\) \(c_7 = 2c_{6} + c_5 = 2 \cdot 70 + 29 = 169\) \(c_8 = 2c_{7} + c_6 = 2 \cdot 169 + 70 = 408\) So, the value of \(c_8\) is 408.

Key Concepts

Recurrence Relation

Understanding recurrence relations is crucial for solving problems in discrete mathematics and computer science. A recurrence relation is an equation that defines a sequence of numbers where each term is a function of the preceding ones. It is very much like a domino effect: once you tap into the first few terms, you can watch the rest of the sequence unfold according to the rule established by the recurrence relation.

For example, if we have a sequence where each term is double the previous term plus the term before that, we can write the recurrence relation as: \( c_n = 2c_{n-1} + c_{n-2} \). We need initial conditions to start the sequence, which in this case could be \( c_0 = 0 \) and \( c_1 = 1 \). With these, the entire sequence can be computed.

Sequence Computation

Once you have a recurrence relation, sequence computation is the process of finding the terms of the sequence. It's essentially building the sequence term by term, based on the initial conditions and the recurrence relation. It can be done manually, which is effective for small sequences, or by using algorithms or programming for larger and more complex ones.

To manually compute a sequence, you start with known terms (like \( c_0 \) and \( c_1 \) in our exercise). Then, one at a time, you calculate each subsequent term using the recurrence formula. You need to be methodical and ensure each step is correct, as any mistake will carry through to the rest of the calculation. This can be a time-consuming process, but it's a good exercise in understanding how the terms of the sequence are dependent on each other.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement is true for all natural numbers. It is divided into two steps. First, the 'base case' where the statement is proven to be true for the initial number, usually \( n = 0 \) or \( n = 1 \). Second, the 'inductive step' where we assume the statement is true for \( n \) and then prove it for \( n + 1 \).

In recurrence relations, mathematical induction can show that a formula holds for all terms of the sequence after we have established the initial terms. The inductive step relies on the assumption that if the relation works for the first few numbers, it will continue to work for subsequent ones. This principle underlies sequence computation as well, ensuring the formula we use yields correct results throughout the sequence.