The given sequence is:

$3, 6, 10, 15$

To identify the pattern and find the next term, we calculate the differences between consecutive terms:

• $6 - 3 = 3$
• $10 - 6 = 4$
• $15 - 10 = 5$

The differences between the terms are increasing by 1 each time (3, 4, 5, ...). This suggests that the sequence is generated by adding consecutive natural numbers to the previous term.

Thus, to find the next term:

• The next difference should be $5 + 1 = 6$.
• Therefore, the next term is $15 + 6 = 21$.

The next term in the sequence is 21.

This sequence follows a pattern where each term can be expressed as a triangular number, which is calculated as:

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

For example:

• $T_2 = \frac{2(2+1)}{2} = 3$
• $T_3 = \frac{3(3+1)}{2} = 6$
• $T_4 = \frac{4(4+1)}{2} = 10$
• $T_5 = \frac{5(5+1)}{2} = 15$
• $T_6 = \frac{6(6+1)}{2} = 21$

Would you like more details or have any questions?

Related Questions:

1. What is the 10th term of this sequence?
2. How can we derive a general formula for the nth term of a triangular number sequence?
3. What is the sum of the first 10 terms of this sequence?
4. How does this sequence relate to Pascal's Triangle?
5. How are triangular numbers used in real-life applications?

Tip:

Triangular numbers can be visualized as dots forming an equilateral triangle, which helps in understanding their geometric properties.