What are the next patterns in the following sequences, given the number of circles in each pattern?

The sequence is 1, 3, 6, 10.

This is a sequence of triangular numbers, where the $n$-th term is $T_n = \frac{n(n+1)}{2}$.
The given patterns correspond to $T_1, T_2, T_3, T_4$.

The next term is $T_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = 15$.

The next pattern will have 15 circles arranged in a triangle with 5 circles at the base.

2.

The sequence is 3, 6, 10, 15.

This is a sequence of triangular numbers, starting from $T_2$. The given patterns correspond to $T_2, T_3, T_4, T_5$.

The next term is $T_6 = \frac{6(6+1)}{2} = \frac{6 \times 7}{2} = 21$.

The next pattern will have 21 circles arranged in a triangle with 6 circles at the base.

3.

The sequence is 1, 3, 6, 10.

This is a sequence of triangular numbers, where the $n$-th term is $T_n = \frac{n(n+1)}{2}$.
The given patterns correspond to $T_1, T_2, T_3, T_4$.

The next term is $T_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = 15$.

The next pattern will have 15 circles arranged in a staircase shape, with 5 circles in the bottom row, 4 in the row above it, 3, 2, and 1 at the top.

4.

The sequence is 1, 3, 6, 10.

This is a sequence of triangular numbers, where the $n$-th term is $T_n = \frac{n(n+1)}{2}$.
The given patterns correspond to $T_1, T_2, T_3, T_4$, representing a triangle with $n$ dots on each side.

The next term is $T_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = 15$.

The next pattern will have 15 dots. It will form a large triangle with 5 dots on each side, and all dots connected by lines to form smaller triangles within.