🟣 Special Sequences

In this topic, we'll delve into three iconic examples of special sequences:

• The Fibonacci sequence: A sequence where each number is the sum of the two preceding ones. It's famous for its presence in nature and art.
• Square numbers: The sequence of numbers that are perfect squares (1, 4, 9, 16, ...). They represent geometric squares and have fundamental properties in number theory.
• Triangular numbers: Numbers that represent the count of objects arranged in an equilateral triangle (1, 3, 6, 10, ...). They have connections to combinatorics and geometry.

The Fibonacci sequence is one of the most famous and fascinating sequences in mathematics. It starts with 0 and 1, and each subsequent number is the sum of the two numbers immediately before it.

The sequence begins:

$$\{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots\}$$

Construction

Let's see how it's constructed:

• The first term, $F_0 = 0$
• The second term, $F_1 = 1$
• From the third term onwards, each term is the sum of the previous two:
  • $F_2 = F_1 + F_0 = 1 + 0 = 1$
  • $F_3 = F_2 + F_1 = 1 + 1 = 2$
  • $F_4 = F_3 + F_2 = 2 + 1 = 3$
  • $F_5 = F_4 + F_3 = 3 + 2 = 5$
  • and so on...

Recursive Formula

We can define the Fibonacci sequence using a recursive formula:

$$F_0 = 0, \quad F_1 = 1, \quad F_n = F_{n-1} + F_{n-2} \text{ for } n \geq 2$$

This formula tells us how to find any term in the sequence if we know the two terms before it.

The Fibonacci sequence is not just a mathematical construct; it appears remarkably often in nature and various other fields:

Nature

• Flower petals: The number of petals in many flowers is a Fibonacci number (e.g., lilies have 3, buttercups have 5, daisies often have 34 or 55)
• Spiral arrangements: The spirals in sunflowers, pine cones, and seed heads often follow Fibonacci numbers in their counts
• Branching of trees: The way branches split off from tree trunks sometimes follows Fibonacci patterns
• Shells: The nautilus shell grows in a Fibonacci spiral

Art and Architecture

The Golden Ratio (approximately 1.618), which is closely related to the Fibonacci sequence, has been used by artists and architects for centuries to create aesthetically pleasing proportions. As you go further in the Fibonacci sequence, the ratio of consecutive terms approaches the Golden Ratio.

$$\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$

Computer Science

The Fibonacci sequence appears in algorithms, data structures, and analysis of computational processes. For example, Fibonacci heaps and Fibonacci search techniques are named after this sequence.

Example 1: Finding $F_6$

To find a specific Fibonacci number, we can use the recursive definition. Let's find the 6th Fibonacci number ($F_6$):

We know: $F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5$

Then, using the formula $F_n = F_{n-1} + F_{n-2}$:

$$F_6 = F_5 + F_4 = 5 + 3 = 8$$

Answer: The 6th Fibonacci number ($F_6$) is 8.

Example 2: Finding $F_{10}$

Find the 10th Fibonacci number.

Solution: Build up from what we know:

• $F_6 = 8$ (from Example 1)
• $F_7 = F_6 + F_5 = 8 + 5 = 13$
• $F_8 = F_7 + F_6 = 13 + 8 = 21$
• $F_9 = F_8 + F_7 = 21 + 13 = 34$
• $F_{10} = F_9 + F_8 = 34 + 21 = 55$

Answer: $F_{10} = 55$

Example 3: Fibonacci in Pascal's Triangle

The Fibonacci numbers can be found by summing diagonals in Pascal's Triangle! This is a beautiful connection between two famous mathematical structures.

Square numbers are sequences of numbers that represent the area of squares with integer side lengths. They are obtained by squaring the natural numbers (1, 2, 3, 4, ...).

The sequence of square numbers starts:

$$\{1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots\}$$

Each term in the sequence is a perfect square:

• The first square number is $1^2 = 1$
• The second square number is $2^2 = 4$
• The third square number is $3^2 = 9$
• The fourth square number is $4^2 = 16$
• and so on...

Formula

The formula for the n-th square number, often denoted as $S_n$ or $Sq_n$, is simply:

$$S_n = n^2$$

Where $n$ starts from 1 (for the first term), 2 (for the second term), 3, and so forth.

Square numbers are naturally visualized as square arrangements of dots or units.

• 1st square number (1): One dot forms a 1×1 square
• 2nd square number (4): Four dots arrange into a 2×2 square
• 3rd square number (9): Nine dots form a 3×3 square
• 4th square number (16): Sixteen dots make a 4×4 square
• And so on...

Square numbers have several interesting properties in mathematics:

• Sum of odd numbers: The sum of the first $n$ odd numbers is equal to the n-th square number. For example: $1 = 1^2$, $1 + 3 = 4 = 2^2$, $1 + 3 + 5 = 9 = 3^2$, $1 + 3 + 5 + 7 = 16 = 4^2$
• Consecutive differences: The difference between consecutive square numbers increases by 2 each time. For example: $4 - 1 = 3$, $9 - 4 = 5$, $16 - 9 = 7$, $25 - 16 = 9$. The differences form the arithmetic sequence $\{3, 5, 7, 9, \ldots\}$
• Fundamental in mathematics: Square numbers are fundamental in number theory, algebra, and geometry

Example 4: Finding the 10th Square Number

To find the 10th square number, we simply use the formula $S_n = n^2$ with $n = 10$.

$$S_{10} = 10^2 = 100$$

Answer: The 10th square number is 100.

Example 5: Sum of First n Odd Numbers

Verify that $1 + 3 + 5 + 7 + 9 = 5^2$

Solution:

$1 + 3 + 5 + 7 + 9 = 25 = 5^2$ ✓

This confirms that the sum of the first 5 odd numbers equals the 5th square number!

Example 6: Finding Which Square Number

Is 144 a square number? If yes, which one?

Solution:

We need to find if there exists an integer $n$ such that $n^2 = 144$

$\sqrt{144} = 12$

Since 12 is an integer, 144 is indeed a square number.

Answer: Yes, 144 is the 12th square number ($12^2 = 144$)

Triangular numbers are sequences of numbers that represent the total count of objects arranged in equilateral triangles with increasing side lengths.

The sequence of triangular numbers begins:

$$\{1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots\}$$

Visual Formation

Let's see how these numbers are formed visually:

• 1st triangular number (1): A single dot forms a triangle
• 2nd triangular number (3): Three dots form a triangle with base 2
• 3rd triangular number (6): Six dots form a triangle with base 3
• 4th triangular number (10): Ten dots make a triangle with base 4
• And so on...

The formula for the n-th triangular number, often denoted as $T_n$ or $Tr_n$, is given by:

$$T_n = \frac{n(n + 1)}{2}$$

This formula is derived from the sum of the first $n$ natural numbers ($1 + 2 + 3 + \ldots + n$). In fact, the n-th triangular number is equal to the sum of the first $n$ positive integers.

$$T_n = 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2}$$

Triangular numbers also have interesting properties and connections:

• Connection to square numbers: The sum of two consecutive triangular numbers is always a square number. For example: $T_1 + T_2 = 1 + 3 = 4 = 2^2$, $T_2 + T_3 = 3 + 6 = 9 = 3^2$, $T_3 + T_4 = 6 + 10 = 16 = 4^2$
• Every square from consecutive triangulars: Every square number is the sum of two consecutive triangular numbers
• Combinatorics connection: Triangular numbers appear in combinatorics, such as in counting the number of ways to choose 2 items from a set of $n+1$ items (combinations): $\binom{n+1}{2} = T_n$
• Handshake problem: If $n$ people each shake hands with every other person exactly once, the total number of handshakes is $T_{n-1}$

Example 7: Finding the 8th Triangular Number

To find the 8th triangular number, we use the formula $T_n = \frac{n(n + 1)}{2}$ with $n = 8$.

$$T_8 = \frac{8(8 + 1)}{2} = \frac{8 \times 9}{2} = \frac{72}{2} = 36$$

Answer: The 8th triangular number is 36.

Example 8: Verifying the Square Number Property

Verify that $T_5 + T_6$ is a square number.

Solution:

First, find $T_5$ and $T_6$:

$T_5 = \frac{5(6)}{2} = 15$

$T_6 = \frac{6(7)}{2} = 21$

Now add them:

$T_5 + T_6 = 15 + 21 = 36 = 6^2$ ✓

Answer: Yes, their sum is 36, which is the square number $6^2$!

Example 9: The Handshake Problem

If 10 people are in a room and each person shakes hands with every other person exactly once, how many handshakes occur?

Solution:

This is the 9th triangular number (since with 10 people, we have $n-1 = 9$):

$T_9 = \frac{9(10)}{2} = 45$

Answer: There are 45 handshakes.