How can I explain how I figured out the 99th Triangular Number?

I'd appreciate it if it is as simple as possible.

Did you figure it out? What did you get? Are you referring to the series of numbers 1,3,6, 10, 14 etc?

You might examine the initial sequence of triangular numbers to if they had some property that would allow you to derive the nth triangular number?

n.....1.....2.....3.....4.....5......6...
N...1.....3.....6....10...15....21...
DIff...2.....3.....4......5.....6
Diff......1......1......1.....1

Since the second differences are equal, the sequence is a finite difference sequence making the nth term of the form an^2 + bn + c = 0

Using the data derived:
1--a(1)^2 + b(1) + c = 1 or a + b + c = 1
2--a(2)^2 + b(2) + c = 3 or 4a + 2b + c = 3
3--a(3)^2 + b(3) + c = 6 or 9a + 3b + c = 6
4--Subtracting (2) from (1) yields 3a + b = 2
5--Subtracting (3) from (2) yields 5a + b = 3
6--Subtracting (4) from (5) yields 2a = 1 making a = 1/2
7--Substituting a = 1/2 back into (2) yields b = 1/2
8--From (1), c = 0

Therefore, the nth term, Nn = n^2/2 + n/2 + 0 = (n^2 + n)/2 = n(n + 1)/2.

To calculate the 99th Triangular Number, you can use a formula or a simple calculation method. Here's a step-by-step explanation of how to figure it out:

1. Recall that a Triangular Number is the sum of all the natural numbers from 1 up to a given number. For example, the 5th Triangular Number is 1+2+3+4+5 = 15.

2. The formula to find the nth Triangular Number is: Tn = n(n+1)/2, where n represents the position of the Triangular Number in the sequence.

3. In this case, you're looking for the 99th Triangular Number. Plug in the value of n into the formula: T99 = 99(99+1)/2.

4. Simplify the formula using basic arithmetic operations. Multiply 99 and 100 which equals 9900, then divide by 2 to get 4950.

Therefore, the 99th Triangular Number is 4950. You can explain this by saying that you used the formula Tn = n(n+1)/2 and substituted 99 for n to get the solution 4950.