Problem 61 - Thinking Mathematically

Question

The ancient Greeks studied figurate numbers, so named because of their representations as geometric arrangements of points. a. Use inductive reasoning to write the five triangular numbers that follow \(21 .\) b. Use inductive reasoning to write the five square numbers that follow \(25 .\) c. Use inductive reasoning to write the five pentagonal numbers that follow 22 . d. Use inductive reasoning to complete this statement: If a triangular number is multiplied by 8 and then 1 is added to the product, a number is obtained.

Step-by-Step Solution

Answer

a. The next five triangular numbers are 28, 36, 45, 55, and 66.\n b. The next five square numbers are 36, 49, 64, 81, and 100.\n c. The next five pentagonal numbers are 35, 51, 70, 92, and 117.\n d. If a triangular number is multiplied by 8 and then 1 is added to the product, a number that fits the formula \(8N + 1\) is obtained.

Step 1: a. Triangular Numbers

Triangular numbers are a sequence where each number is the sum of all natural numbers up to a certain point. Following \(21\), the next five triangular numbers can be obtained by adding the next natural number to the previous triangular number. So, \(21 (6^{th}) + 7 = 28 (7^{th})\), \(28 + 8 = 36 (8^{th})\), \(36 + 9 = 45 (9^{th})\), \(45 + 10 = 55 (10^{th})\), \(55 + 11 = 66 (11^{th})\).

Step 2: b. Square Numbers

Square numbers are essentially perfect squares, i.e., the product of some integer with itself. Following 25, we can find the next five square numbers by squaring the next five natural numbers. So, \(6^2 = 36\), \(7^2 = 49\), \(8^2 = 64\), \(9^2 = 81\), \(10^2 = 100\).

Step 3: c. Pentagonal Numbers

Pentagonal numbers are generated by the formula \(P_n = \frac{n(3n-1)}{2}\) for \(n >= 1\). To find the five pentagonal numbers that follow 22, increment n by 1 for each subsequent pentagonal number. So, \(P_5 = \frac{5(3*5-1)}{2} = 35\), \(P_6 = \frac{6(3*6-1)}{2} = 51\), \(P_7 = \frac{7(3*7-1)}{2} = 70\), \(P_8 = \frac{8(3*8-1)}{2} = 92\), \(P_9 = \frac{9(3*9-1)}{2} = 117\).

Step 4: d. Multiplication and Addition on Triangular Numbers

The statement implies that by multiplying a triangular number by 8 and adding 1, a certain number is obtained. This can be written as \(8N + 1\), where N is a triangular number. This pattern holds true for all triangular numbers.

Key Concepts

Inductive Reasoning

Inductive reasoning is a logical process in which multiple premises, all believed to be true, are combined to obtain a specific conclusion. It is often used in mathematics to predict outcomes and identify patterns in a sequence. For example, to continue a sequence of numbers with a certain pattern, we would observe the existing elements of the sequence, establish a relationship, and apply that generalized rule to predict the following numbers. In our exercise, inductive reasoning is used to derive subsequent figurate numbers by analyzing the given numbers and inferring the rules that govern the progression. It's the detective work of mathematics; you spot clues, formulate a hypothesis, and test it with each new piece of evidence.

Triangular Numbers

Triangular numbers are named so because they can be represented as a triangle of dots. The nth triangular number is found by adding up the natural numbers up to and including n. This can be expressed as \( T_n = \frac{n(n + 1)}{2} \). By adding the next consecutive natural number to the previous triangular number or using the formula, we can find subsequent triangular numbers.

By applying this method, after the 6th triangular number, which is 21, the next five triangular numbers can be calculated. It is like stacking dots in a triangle, layer by layer, with each layer having one more dot than the layer above it. It's essential for students to visualize this geometric pattern to fully comprehend how triangular numbers grow.

Square Numbers

Square numbers, also known as perfect squares, are integers that are the square of an integer. In other words, a number is a square number if it can be written as \( n^2 \) where n is an integer. Visually, these numbers can be arranged to form a perfect square.

The sequence of square numbers begins with 1, 4, 9, 16, and so forth since these are the squares of the natural numbers starting from 1. After exploring the sequence, students can easily predict that after the fifth square number (25), the next five would be 36, 49, 64, 81, and 100, aligning with the squares of the numbers from 6 to 10. Encouraging the visualization of these as squares on a grid helps students to grasp the concept quicker.

Pentagonal Numbers

Pentagonal numbers are a fascinating type of figurate numbers that can form a pentagon when dots representing them are placed in a certain pattern. The formula used to calculate the nth pentagonal number is \( P_n = \frac{n(3n - 1)}{2} \).

To find the pentagonal numbers that follow a given one, we use the formula and sequentially increase n. The beauty of pentagonal numbers is in their arrangement. One can imagine dots being added in layers to form a larger pentagon each time. For example, proceeding from the pentagonal number 22, we increase the n value appropriately to discover the subsequent ones: 35, 51, 70, 92, and 117. This concept introduces geometric growth in an easily visual format, which is integral for students' understanding of this elegant mathematical concept.