Problem 20 - Discrete Mathematics with Applications

Question

The nth Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) h 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. $$ F_{1} $$

Step-by-Step Solution

Answer

In order to compute the Fibonacci number \(F_1\), we plug in \(n=1\) into the given formula, which results in a sum that has only one term with \(i=0\). Evaluating the binomial coefficient, we get \(\left(\begin{array}{c}{0}\\{0}\end{array}\right)=1\). Thus, the first Fibonacci number, \(F_{1}\), is equal to 1.

Step 1: Plug in the value of n into the formula

First, let's plug in the value of \(n=1\) into the formula: $$ F_{1}=\sum_{i=0}^{\lfloor(1-1)/2\rfloor}\left(\begin{array}{c}{1-i-1}\\{i}\end{array}\right) $$

Step 2: Evaluate the expression inside the floor function

Before we can find the sum, let's evaluate the expression inside the floor function: $$ \lfloor (1-1)/2 \rfloor = \lfloor 0/2 \rfloor = \lfloor 0 \rfloor = 0 $$

Step 3: Find the limits of the sum

Now that we have the value of the floor function, we can find the limits of our sum: $$ F_{1}=\sum_{i=0}^{0}\left(\begin{array}{c}{1-i-1}\\{i}\end{array}\right) $$ Since the sum starts and ends with \(i=0\), there is only one term to evaluate.

Step 4: Evaluate the binomial coefficient

Let's calculate the binomial coefficient for the given term, where \(i=0\): $$ \left(\begin{array}{c}{1-0-1}\\{0}\end{array}\right) = \left(\begin{array}{c}{0}\\{0}\end{array}\right) $$ By definition, the binomial coefficient \(\left(\begin{array}{c}{0}\\{0}\end{array}\right) = 1\).

Step 5: Determine the value of F1

Now that we have the value of the binomial coefficient, we can determine the value of \(F_{1}\): $$ F_{1}=\sum_{i=0}^{0}\left(\begin{array}{c}{1-i-1}\\{i}\end{array}\right) = \left(\begin{array}{c}{0}\\{0}\end{array}\right) = 1 $$ So, the first Fibonacci number, \(F_{1}\), is equal to 1.

Key Concepts

Pascal's triangle

Pascal's triangle is a fascinating mathematical structure that displays a triangular arrangement of binomial coefficients. It is named after the French mathematician Blaise Pascal, though it was studied centuries earlier in other cultures.
At its most basic, each number in Pascal's triangle is the sum of the two numbers directly above it in the previous row. The rows of Pascal's triangle represent the coefficients in the expansion of binomial expressions such as \((a + b)^n\).
Pascal's triangle is much more than a simple arrangement of numbers. It has numerous properties and connections to other areas of mathematics, such as:

• Combinations and binomial coefficients
• Symmetry across its center
• Connections to the Fibonacci sequence

These connections make Pascal's triangle a powerful tool in combinatorics and algebra.

Binomial coefficient

A binomial coefficient is a specific type of coefficient that appears in the expansion of binomials. It is typically represented through the notation \(\binom{n}{k}\), which is read as "n choose k."
This notation represents the number of ways to choose k elements from a set of n distinct elements without regard to order. Essentially, it's about counting possibilities.
The formula for calculating a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• \(n!\) is the factorial of n (i.e., the product of all positive integers up to n).
• \(k!\) is the factorial of k.
• \((n-k)!\) is the factorial of the difference between n and k.

Binomial coefficients arise naturally in many areas of mathematics, especially in combinatorics and algebra, and are critical components in Pascal's triangle.

Mathematical proof

Mathematical proof is a logical argument demonstrating the truth of a statement in mathematics. Proofs are essential in mathematics because they provide verification that assertions about mathematical objects hold under specified conditions. There is a wide range of proof techniques, including:

• Direct proof
• Indirect proof
• Proof by induction
• Proof by contradiction
• Proof by contrapositive

Each method relies on a clear understanding of logical reasoning and accepted mathematical truths.
Using proof ensures that results are not just accidental or coincidental, but genuinely hold in all specified cases. Through crafting a proof, often mathematicians explore deeper insights into problems, revealing new connections and understandings. Proofs instill confidence in results, allowing them to be used reliably in further mathematical work.

Floor function

The floor function is a mathematical function that outputs the greatest integer less than or equal to a given number. Denoted as \(\lfloor x \rfloor\), it effectively "rounds down" a number to the nearest integer.
For instance, \(\lfloor 3.7 \rfloor = 3\), and \(\lfloor -2.5 \rfloor = -3\). The floor function is crucial in various calculations, especially when dealing with discrete data in computational mathematics.
Key properties of the floor function include:

• If \(x\) is an integer, then \(\lfloor x \rfloor = x\).
• For any real number \(x\), \(x - 1 < \lfloor x \rfloor \leq x\).
• It is a piecewise constant function that is continuous from the right.

The floor function is especially useful when performing mathematical operations involving division and when ensuring results are integers. It plays a critical role in formulations and algorithms where rounding and integer calculations are needed.