Hey my question is

How do we get the triangular numbers sequence ( general statement ) by the quadratic formula ?
Can you shoe me steps please ?
The triangle numbers are :
1 , 3 , 6 , 10 , 15

how do I get them in the form of a quadratic equation if a , b and c are unknown ?
Plz help with steps :(

To find the general statement for the triangular numbers sequence using the quadratic formula, we can start by analyzing the differences between consecutive terms. Let's use the triangular numbers sequence you provided as an example: 1, 3, 6, 10, 15.

Step 1: Find the first differences.
The first differences are obtained by subtracting each term from the next term:
3 - 1 = 2
6 - 3 = 3
10 - 6 = 4
15 - 10 = 5

Step 2: Find the second differences.
The second differences are obtained by subtracting each first difference from the next first difference:
3 - 2 = 1
4 - 3 = 1
5 - 4 = 1

Step 3: Check if the second differences are constant.
In this case, the second differences obtained in Step 2 are all equal to 1. This indicates that the original sequence follows a quadratic pattern.

Step 4: Set up the quadratic equation.
Let's assume the general quadratic equation is of the form: f(n) = an^2 + bn + c, where f(n) represents the nth term in the sequence.

Step 5: Use the first term to find the value of 'c'.
Since the first term of the sequence is 1, we can substitute n = 1 into the quadratic equation and solve for 'c':
1 = a(1)^2 + b(1) + c
1 = a + b + c

Step 6: Use the second term to find the value of 'b'.
Substitute n = 2 into the quadratic equation and use the second term of the sequence (3) to find 'b':
3 = a(2)^2 + b(2) + c
3 = 4a + 2b + c

Step 7: Use the third term to find the value of 'a'.
Substitute n = 3 into the quadratic equation and use the third term of the sequence (6) to solve for 'a':
6 = a(3)^2 + b(3) + c
6 = 9a + 3b + c

Step 8: Solve the system of equations.
Now, we have three equations:
1 = a + b + c
3 = 4a + 2b + c
6 = 9a + 3b + c

Solving this system of equations will give us the values of 'a', 'b', and 'c'.

Step 9: Substitute the values of 'a', 'b', and 'c' back into the quadratic equation.
Once you have the values of 'a', 'b', and 'c', substitute them back into the quadratic equation: f(n) = an^2 + bn + c.

By following these steps and solving the system of equations, you can obtain the general quadratic equation that represents the triangular numbers sequence.