Problem 21 - 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. $$F_{2}$$

Step-by-Step Solution

Answer

In this problem, we are asked to compute the Fibonacci number at position 2 (\(F_{2}\)) using the given formula. Substituting \(n = 2\) into the formula and simplifying, we get: $$F_{2}=\sum_{i=0}^{0}\left(\begin{array}{c} 2-i-1 \\ i \end{array}\right)$$ Since our sum only has one term (i = 0), we can compute the value inside the sum: $$F_{2}=\left(\begin{array}{c} 1 \\\ 0 \end{array}\right)$$ From Pascal's triangle, we find the value at row 1 and column 0 to be 1. Thus, using the given formula, the Fibonacci number at position 2 is \(F_{2}=1\).

Step 1: Understand the given formula

To calculate the nth Fibonacci number, the given formula is: $$F_{n}=\sum_{i=0}^{\lfloor(n-1) / 2\rfloor}\left(\begin{array}{c} n-i-1 \\\ i \end{array}\right)$$ In this formula, n is the position of the Fibonacci number in the sequence. We need to compute \(F_{2}\), so we'll replace \(n\) with 2 in the given formula.

Step 2: Substitute n with 2 in the formula

Now, let's replace \(n\) with 2 in the given formula: $$F_{2}=\sum_{i=0}^{\lfloor(2-1) / 2\rfloor}\left(\begin{array}{c} 2-i-1 \\\ i \end{array}\right)$$

Step 3: Simplify the formula

Let's simplify the formula for \(F_{2}\): 1. Calculate the value inside the floor function: \(\lfloor(2-1) / 2\rfloor= \lfloor(1) / 2\rfloor= \lfloor0.5\rfloor= 0\) 2. Substitute the value inside the sum: $$F_{2}=\sum_{i=0}^{0}\left(\begin{array}{c} 2-i-1 \\\ i \end{array}\right)$$

Step 4: Compute the sum

Now that our sum only has one term (i = 0), we can compute the value inside the sum: $$F_{2}=\left(\begin{array}{c} 2-0-1 \\ 0 \end{array}\right)=\left(\begin{array}{c} 1 \\\ 0 \end{array}\right)$$ Looking at Pascal's triangle, we can see that the value at row 1 and column 0 is 1. So, we have: $$F_{2}=1$$

Step 5: Final answer

The Fibonacci number at position 2, using the given formula, is: $$F_{2}=1$$

Key Concepts

Pascal's triangle

Pascal's triangle is a triangular array where each row represents the coefficients of the binomial expansion. It's an important tool in many areas of mathematics due to its connection to binomial coefficients and combinatorial identities. In each row, the number above and to the left and the number above and to the right are added together to form the number below.

• Row 0 contains a single 1.
• Each subsequent row starts and ends with 1.
• Each interior number is the sum of the two numbers directly above it in the previous row.

Pascal's triangle is uniquely structured so that the nth row corresponds to the coefficients of \((a + b)^n\) when expanded. These coefficients are also known as binomial coefficients. Observing the triangle provides insight into the relationship between combinations and algebraic expansions. Since Fibonacci numbers can also be derived from Pascal's triangle, it illustrates the deep connection between different mathematical domains.

binomial coefficients

Binomial coefficients are the numbers that appear as coefficients in the binomial theorem, given by the expression \((a + b)^n\). The formula for a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(n!\) indicates the factorial of \(n\). These coefficients can be calculated directly from this formula or by using Pascal's triangle.

• Expressed as \(C(n, k)\) or \(\binom{n}{k}\).
• They enumerate ways to choose \(k\) objects from \(n\) objects without considering the order.

Binomial coefficients play a crucial role in combinatorics, algebra, and probability theory. They are essential in calculating combinations, which is often required in solving problems involving permutations and combinations. Additionally, they naturally appear when expanding powers of binomials into polynomials.

combinatorial identities

Combinatorial identities are equations that involve combinatorial quantities like binomial coefficients. These identities allow us to understand and solve complex combinatorial problems. They often facilitate simplifications and multiplications in combinatorial proofs or arguments.

• One fundamental identity is Pascal's rule: \( \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1} \)
• Other useful ones include symmetry, \(\binom{n}{k} = \binom{n}{n-k}\), and the binomial theorem itself.

Each identity reveals the manner in which combinatorial quantities relate to each other. They are critical to simplifying expressions in proofs and derive naturally from properties of the binomial coefficients in Pascal’s triangle. These identities have applications in algorithm design, scientific computation, and various fields that utilize discrete mathematics.