To find the next triangular number in the sequence following 78 and 91, we need to understand the formula for triangular numbers.

The formula for the $n$-th triangular number $T_n$ is:

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

We are given two consecutive triangular numbers: 78 and 91. Let's determine their positions in the sequence.

Step 1: Identify the position of 78 and 91

We set up the equation for 78:

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

Multiplying both sides by 2:

$n(n + 1) = 156$

This is a quadratic equation:

$n^2 + n - 156 = 0$

Solving this quadratic using the quadratic formula:

$n = \frac{-1 \pm \sqrt{1^2 + 4 \cdot 1 \cdot 156}}{2 \cdot 1}$ $n = \frac{-1 \pm \sqrt{1 + 624}}{2}$ $n = \frac{-1 \pm \sqrt{625}}{2}$ $n = \frac{-1 \pm 25}{2}$

Taking the positive root:

$n = \frac{-1 + 25}{2} = 12$

Thus, 78 is the 12th triangular number.

Similarly, check for 91:

$T_{13} = \frac{13(13 + 1)}{2} = \frac{13 \times 14}{2} = 91$

So, 91 is the 13th triangular number.

Step 2: Find the next triangular number

The next triangular number after $T_{13} = 91$ is $T_{14}$.

$T_{14} = \frac{14(14 + 1)}{2} = \frac{14 \times 15}{2} = \frac{210}{2} = 105$

Final Answer:

The next triangular number after 91 is:

$\boxed{105}$

Let me know if you have any questions or would like further details.

Follow-Up Questions:

1. What is the formula for the $n$-th triangular number?
2. How can you determine the position of a given triangular number in the sequence?
3. What are the first 10 triangular numbers?
4. How can triangular numbers be used in solving problems involving arrangements of objects?
5. Can you express the relationship between triangular numbers and square numbers?

Tip:

Triangular numbers can be visualized as dots arranged in the shape of an equilateral triangle, which can help you understand their growth pattern and recognize them in sequences.