1, 3, 6, 10, 15, ... what comes next?

Question

Look at the sequence: 1,\ 3,\ 6,\ 10,\ 15,\ \dots What is the next number? And can you jump straight to the 100th number in this list without writing out all the others?

Accepted Answer

The next number is 21. Look at the gaps between terms. Write the differences from one term to the next: 1 \xrightarrow{+2} 3 \xrightarrow{+3} 6 \xrightarrow{+4} 10 \xrightarrow{+5} 15. The amount added each step grows by one: +2, +3, +4, +5. So the next step adds +6, giving 15 + 6 = \mathbf{21}. What these numbers are. These are the triangular numbers, the nth one is the sum 1 + 2 + 3 + \cdots + n, the number of dots in a triangle with n dots along each side. The 5th term is 1+2+3+4+5 = 15, and the 6th is 15 + 6 = 21. The shortcut to the 100th term. There's a neat closed form: 1 + 2 + \cdots …

1, 3, 6, 10, 15, ... what comes next?

Your answer

More Brainteasers questions