When objects are arranged in certain ways, they form a pattern. For example, square numbers can be obtained by forming objects in square arrangements. As shown below, the first four square numbers are 1, 4, 9, 16 and 25.
Clearly, the number of dots in the 100th square number is . In general, the formula for finding the nth square number is .
Now, how do we extend the formula of polygonal numbers? For example, what is the 6th pentagonal number (shown in the figure above) and the 10th heptagonal number? Try to see if you can see a pattern from the second figure. If you can derive the formula for the triangular numbers, it will be easier for you to solve the problem. You can also answer the following questions.
- What is the 3rd hexagonal number?
- What is the 5th octagonal number?
- What is the 2nd heptagonal number?
The image above is an example of proof without words. Proof without words (well, they are not really proof per se ) are images or statements that can help the reader see if a given statement is true or false even without accompanying explanations. We will have more of this in the posts future posts.
If you are wondering how to get the answer of the question above, a detailed post of the derivation of the formula for polygonal numbers can be found in Math and Multimedia. However, I suggest that you try your best first before reading the article.
Images via Math and Multimedia and Wikipedia