Problem 23 - Discrete Mathematics with Applications

Question

The \(n\) th Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) th northeast diagonal of Pascal's triangle; that is, $$F_{n}=\sum_{i=0}^{\lfloor(n-1) / 2\rfloor}\left(\begin{array}{c} n-i-1 \\ i \end{array}\right)$$ Using this formula, compute each Fibonacci number. $$F_{6}$$

Step-by-Step Solution

Answer

Using the given formula and Pascal's triangle, we find the coefficients: \[\left(\begin{array}{c} 5 \\ 0 \end{array}\right) = 1, \quad \left(\begin{array}{c} 4 \\ 1 \end{array}\right) = 4, \quad \left(\begin{array}{c} 3 \\ 2 \end{array}\right) = 3\] Then, sum the coefficients to find the 6th Fibonacci number: \(F_{6} = 1 + 4 + 3 = 8\).

Step 1: Identify the values for n and the limits of the sum

We are given n = 6, so we need to find F_6. We have to find the sum from i = 0 to i = floor((n-1)/2). For n = 6, we get floor((6-1)/2) = floor(5/2) = 2.

Step 2: Implement the formula for each i in the summation range

Now we have to find the sum for all i values in the range from 0 to 2. We know the formula is: \[F_{n}=\sum_{i=0}^{\lfloor(n-1) / 2\rfloor}\left(\begin{array}{c} n-i-1 \\\ i \end{array}\right)\] For n = 6, we have to find three elements to sum: \[\left(\begin{array}{c} 6-0-1 \\ 0 \end{array}\right), \quad \left(\begin{array}{c} 6-1-1 \\ 1 \end{array}\right), \quad \left(\begin{array}{c} 6-2-1 \\ 2 \end{array}\right)\]

Step 3: Compute the binomial coefficients

We need to find three binomial coefficients: \[\left(\begin{array}{c} 5 \\ 0 \end{array}\right) = 1, \quad \left(\begin{array}{c} 4 \\ 1 \end{array}\right) = 4, \quad \left(\begin{array}{c} 3 \\ 2 \end{array}\right) = 3\]

Step 4: Sum the coefficients to find the Fibonacci number

Now, sum these coefficients to get the value of F_6: \[F_{6} = 1 + 4 + 3 = 8\] Therefore, the 6th Fibonacci number is \(F_{6} = 8\).

Key Concepts

Pascal's Triangle

Pascal's Triangle is a fascinating mathematical structure named after the French mathematician Blaise Pascal. It's a triangular array of numbers, where each number is the sum of the two directly above it in the previous row. This array starts with a single "1" at the top and grows with each new row.

Each row of Pascal's Triangle corresponds to the coefficients found in the binomial expansion. The \(n\)th row consists of numbers that are the coefficients of \((a + b)^{n}\).

An interesting property of Pascal's Triangle is its connection to different mathematical concepts, such as Fibonacci numbers. By tracing along the diagonal lines of the triangle, we can derive Fibonacci numbers. For instance, if you sum the elements along the northeast diagonals, you will get the Fibonacci sequence.

• The first few rows begin as 1, 1 1, 1 2 1, 1 3 3 1, and so on.
• These numbers progress in a pattern of symmetry and growth.

Binomial Coefficients

Binomial coefficients are an essential part of understanding combinations in mathematics. They are found in Pascal's Triangle and are symbolized as \(\binom{n}{k}\), meaning "\(n\) choose \(k\)." These coefficients represent the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order of selection.

You can calculate them using the formula:
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(!\) denotes factorial, the product of all positive integers up to a number.

These coefficients are crucial in various fields like probability and algebra where they determine the different possibilities and arrangements for a given situation.

• In our exercise, we computed: \(\binom{5}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{3}{2} = 3\).
• Each coefficient corresponds to an element along the Patriot's diagonal, summing to the Fibonacci number.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This branch of mathematics deals with objects that can assume only distinct, separate values. It encompasses a wide range of topics including logic, set theory, graph theory, and number theory.

When we're working with Fibonacci numbers or identifying patterns in Pascal's Triangle, we're diving into areas of discrete mathematics. These patterns enable problem-solving and analytical thinking through well-defined mathematical frameworks.

• Fibonacci sequences represent a classic example in discrete mathematics, illustrating recursive patterns and growth models.
• Understanding binomial coefficients and their applications show how discrete choices add complexity and depth.

This field connects abstract ideas to real-world applications, providing critical thinking skills necessary for computer science, cryptography, and network design.