To predict the next number in the sequence $1, 3, 6, 10, 15$ using inductive reasoning, we can observe the pattern in the differences between consecutive terms.

Let's calculate the differences between each pair of consecutive terms:

$3 - 1 = 2$ $6 - 3 = 3$ $10 - 6 = 4$ $15 - 10 = 5$

The differences between the consecutive terms are $2, 3, 4, 5$, which are increasing by 1 each time.

So, if we follow this pattern, the next difference should be:

$5 + 1 = 6$

To find the next term in the sequence, we add this difference to the last term:

$15 + 6 = 21$

Thus, the next number in the sequence is 21.

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

1. What is inductive reasoning, and how is it used in mathematics?
2. Can you identify a similar pattern in another sequence?
3. How would the sequence change if the pattern of differences changed?
4. What would be the 8th term in this sequence?
5. How does this sequence relate to triangular numbers?

Tip: Recognizing patterns in sequences often involves looking at differences between terms or their ratios.