To find the 7th triangular number after 78, we first need to identify the position of 78 in the sequence of triangular numbers.

The $n$th triangular number is given by the formula:

$T_n = \frac{n(n+1)}{2}$

We need to find $n$ such that:

$\frac{n(n+1)}{2} = 78$

Multiplying both sides by 2:

$n(n+1) = 156$

Now, solve this quadratic equation:

$n^2 + n - 156 = 0$

We can solve this using the quadratic formula:

$n = \frac{-1 \pm \sqrt{1 + 4 \times 156}}{2}$

$n = \frac{-1 \pm \sqrt{625}}{2}$

$n = \frac{-1 \pm 25}{2}$

This gives us:

$n = \frac{24}{2} = 12$

So, 78 is the 12th triangular number.

Now, we want the 7th triangular number after the 12th, which is the 19th triangular number ($12 + 7 = 19$).

To find the 19th triangular number, use the formula again:

$T_{19} = \frac{19 \times 20}{2} = \frac{380}{2} = 190$

So, the 7th triangular number after 78 is 190.

Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the formula to find the $n$th triangular number?
2. How do you derive the quadratic formula?
3. How can triangular numbers be represented geometrically?
4. What is the sum of the first 19 triangular numbers?
5. How are triangular numbers related to square numbers?
6. What is the 10th triangular number after 78?
7. What is the 20th triangular number?
8. Can you derive a general expression for the difference between consecutive triangular numbers?

Tip: Triangular numbers can also be visualized as dots arranged in an equilateral triangle, which can help in understanding their sequence.