What is the next term in the sequence 1, 3, 6, 10?

The next term in the sequence 1, 3, 6, 10 is 15.

This sequence is known as the triangular numbers sequence. Each term in the sequence represents a triangle with dots. For example, the first term is 1, which is just a single dot. The second term is 3, which can be arranged in a triangle with two dots on the bottom row and one dot on top. The third term is 6, forming a triangle with three dots on the bottom row, two in the middle, and one on top, and so on.

To find the next term, you add the next natural number to the last term. The first term is 1. To get the second term, you add 2 to 1, giving you 3. For the third term, you add 3 to the previous term (3), resulting in 6. For the fourth term, you add 4 to the previous term (6), giving you 10. Therefore, to find the fifth term, you add 5 to the previous term (10), which results in 15.

In general, the nth term of the triangular number sequence can be found using the formula: \( T_n = \frac{n(n+1)}{2} \). For example, to find the 5th term, you substitute n with 5: \( T_5 = \frac{5(5+1)}{2} = \frac{30}{2} = 15 \). This confirms that the next term in the sequence is indeed 15.