Question

Devise a formula for the \(n\)th pentagonal number and for the \(n\)th hexagonal number.

Short answer

Answer: The formula for the nth pentagonal number is \(P_n = 1 + (n-1)n\), and the formula for the nth hexagonal number is \(H_n = 1 + (n-1)(2n-3)\).

Step-by-step solution

Analyzing Pentagonal Numbers

To find the \(n\)th pentagonal number, first, let's list the first few pentagonal numbers: 1, 5, 12, 22, 35, ... Now, let's analyze the difference between consecutive pentagonal numbers: 4, 7, 10, 13, ... Notice that the difference between consecutive terms is increasing by 3. Using this information, let's derive the formula for the \(n\)th pentagonal number: 1. 1st term: 1 2. 2nd term: 1 + 4 = 5 3. 3rd term: 1 + 4 + 7 = 12 4. 4th term: 1 + 4 + 7 + 10 = 22 From the pattern, we can deduce that the \(n\)th pentagonal number (\(P_n\)) can be written as: \(P_n = 1 + \big[(1) + (1+3) + (1+3+3) + ... + (1+3+3+...+3(n-2))\big]\) The sum inside the brackets in the above equation is an arithmetic progression with \(n-1\) terms, a starting value of 1, and a common difference of 3.

Deriving the Formula for Pentagonal Numbers

Using the arithmetic progression formula, sum of AP = \(\frac{n-1}{2}[2a + (n-2)d]\), where n is the number of terms, a is the initial term, and d is the common difference. In our case, a=1 and d=3. So, the sum of the arithmetic progression becomes: \(Sum = \frac{n-1}{2}[2(1) + (n-2)(3)] = (n-1)(n-1+1) = (n-1)n\) So, the formula for the \(n\)th pentagonal number (\(P_n\)) is: \(P_n = 1 + (n-1)n\)

Analyzing Hexagonal Numbers

Now, let's find the \(n\)th hexagonal number. First, let's list the first few hexagonal numbers: 1, 6, 15, 28, 45, ... Like before, let's analyze the difference between consecutive hexagonal numbers: 5, 9, 13, 17, ... This time, the difference between consecutive terms is increasing by 4. Using this information, let's derive the formula for the \(n\)th hexagonal number: 1. 1st term: 1 2. 2nd term: 1 + 5 = 6 3. 3rd term: 1 + 5 + 9 = 15 4. 4th term: 1 + 5 + 9 + 13 = 28 From the pattern, we can deduce that the \(n\)th hexagonal number (\(H_n\)) can be written as: \(H_n = 1 + \big[(1+4) + (1+4+4) + (1+4+4+4) + ... + (1+4+4+...+4(n-2))\big]\) The sum inside the brackets in the above equation is also an arithmetic progression with \(n-1\) terms, a starting value of 5 and a common difference of 4.

Deriving the Formula for Hexagonal Numbers

Using the arithmetic progression formula again, sum of AP = \(\frac{n-1}{2}[2a + (n-2)d]\), where n is the number of terms, a is the initial term, and d is the common difference. In our case, a=5 and d=4. So, the sum of the arithmetic progression becomes: \(Sum = \frac{n-1}{2}[2(5) + (n-2)(4)] = (n-1)(2n-3)\) So, the formula for the \(n\)th hexagonal number (\(H_n\)) is: \(H_n = 1 + (n-1)(2n-3)\) In conclusion, the formula for the \(n\)th pentagonal number is \(P_n = 1 + (n-1)n\), and the formula for the \(n\)th hexagonal number is \(H_n = 1 + (n-1)(2n-3)\).

Key concepts

Arithmetic Progression

Arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed value, known as the common difference, to the previous term. Understanding how arithmetic progressions work is essential in deriving formulas for certain number series, like pentagonal and hexagonal numbers.

• Each arithmetic progression is defined by:
  • First term (\(a\)).
  • Common difference (\(d\)).
  • Number of terms (\(n\)).

The sum of an arithmetic sequence can be calculated with the formula:\[S_n = \frac{n}{2} [2a + (n-1)d]\]This formula is useful in finding the sum of the differences between consecutive pentagonal or hexagonal numbers, which follow their unique arithmetic sequences with different common differences.
In pentagonal numbers, for example, the common difference is 3, whereas in hexagonal numbers, it is 4. By identifying these properties, we can derive general formulas for these numbers over the sequence.

Number Theory

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers. It is crucial when analyzing sequences like the pentagonal and hexagonal numbers, as it helps in understanding their structure and formulating their expressions.
Number theory involves studying various kinds of numbers, such as:

• Natural numbers (e.g., 1, 2, 3,...).
• Prime numbers (e.g., 2, 3, 5,...).
• Special figurate numbers, like pentagonal and hexagonal numbers.

Pentagonal numbers, such as 1, 5, 12, and so on, and hexagonal numbers like 1, 6, 15, are examples of polyhedral numbers, a type of figurate number representing points arranged in a geometric shape.
Understanding these numbers involves number theory concepts, as finding the formulas for their series requires determining the relationships between terms, growth patterns, and other intrinsic properties.

Polyhedral Numbers

Polyhedral numbers are a category of figurate numbers that extend into three dimensions, forming shapes like polyhedrons. Among these, pentagonal and hexagonal numbers have special geometric significance.

• Pentagonal numbers represent points forming pentagons, a five-sided polygon.
• Hexagonal numbers represent points forming hexagons, a six-sided polygon.

Each pentagonal or hexagonal number corresponds to a specific geometrical arrangement that grows layer by layer, much like the layers of an onion.
The formula for the pentagonal numbers is derived based on the concept of layering triangles, while hexagonal numbers come from layering hexagons within one another. These formulas can be expressed as:

• Pentagonal number formula: \[P_n = 1 + (n-1)n\]
• Hexagonal number formula: \[H_n = 1 + (n-1)(2n-3)\]

Understanding these concepts helps visualize how these numbers evolve, providing insight into the spatial patterns they create.