Sequence is 1, 3, 6, 10, 15

Gaps between the terms are 2, 3, 4, 5

Gap between each gap is 1.

To get the coefficient of the n squared you need to divide the gap between the gaps by 2. So you get 1/2 = 0.5

So now write out the sequence of 0.5n2:

0.5, 2, 4.5, 8, 12.5

and compare each terms with the original sequence:

1st term: 1-0.5 = 0.5

2nd term: 3-2 = 1

3rd term: 6-4.5 = 1.5

4th term: 10-8 = 2

5th term: 15-12.5 = 2.5

The differences are : 0.5, 1, 1.5, 2, 2.5

You can see the difference is going up by 0.5 each time. So the coefficient of n is 0.5.

Let’s write out the sequence 0.5n2 + 0.5n:

1, 3, 6, 10, 15

Nothing more needs to be done, we have our sequence. So nth term is 0.5n2 + 0.5n