How do we know they are all triangular numbers?

[tidyboar]

Module 3 Week 4 Day 13 Challenge Part 1 Explanation

How do we know that each case is actually a triangular number for every single case and that it's not a coincidence?

[debbie]

@tidyboar This is one way to see it (the way I see it), but not the only way:

We're making cases based on a "ceiling" that we choose for our numbers. There is actually both a "ceiling" and a "floor." The ceiling is fixed for each case, and is equal to the value of $c.$ For each fixed value of $c,$ the floor can be raised up and up until it is the same height as the ceiling. For each level of the floor, we have some corresponding values of $(a, b).$ But as the floor is raised, we get fewer and fewer possible values. In fact, raising the floor by $1$ will decrease the number of values of $(a, b)$ by one. This is how we get the "triangle."

Do you see that in the bottom triangle, the value of $a$ is the floor? In the first row, the "floor" is equal to $0,$ so there are many possibilities (3 of them). In the second row, the "floor" is raised up to $1,$ so there are fewer possible values for $(a,b)$ that go up to our ceiling of $c = 2.$ In the third row, the "floor" is equal to the "ceiling," so there is only one possible value: that value where both $a$ and $b$ are equal to the value of the floor/ceiling.

Sorry if that sounded confusing, and please let me know if you would like more explanation on this topic! Thanks for asking and being curious!

[tidyboar]

sorry i don't see why they're triangular numbers

[debbie]

@tidyboar Triangular numbers are the sum of successive consecutive numbers starting from $1,$ like $\textcolor{red}{1, 3, 6,} \textcolor{red}{10}, \textcolor{red}{15} \text{ and } \textcolor{red}{21}.$

$$\begin{aligned} 1 &= \textcolor{red}{1} \\ 1 + 2 &= \textcolor{red}{3} \\ 1 + 2 + 3 &= \textcolor{red}{6} \\ 1 + 2 + 3 + 4 &= \textcolor{red}{10} \\ 1 + 2 + 3 + 4 + 5 &= \textcolor{red}{15} \\ 1 + 2 + 3 + 4 + 5 + 6 &= \textcolor{red}{21} \\ \end{aligned}$$

They are called triangular numbers because if you were to create a triangle out of rows of dots, where each successive row had one more dot than the previous row, then the arrangement would look like a triangle!

$$\begin{aligned} \bullet &= \textcolor{blue}{1} \\ \bullet \text{ } \text{ }\text{ }\bullet &= \textcolor{blue}{2} \\ \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet &= \textcolor{blue}{3} \\ \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet &= \textcolor{blue}{4} \\ \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet &= \textcolor{blue}{5} \\ \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet \text{ }\text{ }\text{ } \bullet &= \textcolor{blue}{6} \\ \\ \text{ total } = \textcolor{blue}{1 + 2 + 3 + 4 + 5 + 6 =} \textcolor{red}{ 21 } \\ \end{aligned}$$

The above illustrates the sixth triangular number, which is $\textcolor{red}{21}.$

In Module 4 Day 5, Prof. Loh goes over how to add up the sum of numbers in such a sequence, where each term increases by the same amount. Such a sequence is called an arithmetic sequence.

Going back to the video, we see that there were $3$ ways to assign values to $a$ and $b$ for Case 2, where $c = 1.$ Oh, $\textcolor{red}{3}$ is a triangular number, since $1 + 2 = \textcolor{red}{3}!$
 

 

There were $6$ ways to assign values to $a$ and $b$ for Case 3, where $c = 2.$ Wow, $\textcolor{red}{6}$ is also a triangular number since it equals $1 + 2 + 3 = \textcolor{red}{6}!$
 

 

Why does the number of entries take the form $1 + 2 + 3 + \ldots$ anyway...?

As I mentioned before, you can imagine that the value of $b$ for the $c = 2$ case must be less than or equal to $2.$ The number $2$ is like a "ceiling." Neither $a$ nor $b$ can exceed the value of this ceiling. They can be equal to it, but not greater than it. Likewise, the value of $a$ is like a "floor," because it determines all the values that are in the middle that are possible. If the value of $a$ is smaller, we have more possible values of $b$ that work. The result of this is that we get the pattern $1 + 2 + 3 + \dots$.

 

Please let me know if this is still unclear, and I'll be happy to try to elucidate upon it!

[tidyboar]

Thank you I understand now!

[debbie]

@tidyboar