using the nth term formula, find the pattern in the sequence 1, 5, 12, 22, 35

The sequence given is 1, 5, 12, 22, 35.

First, let's find the differences between consecutive terms:

5 - 1 = 4 12 - 5 = 7 22 - 12 = 10 35 - 22 = 13

The differences are not constant, so this is not an arithmetic sequence.

Let's find the second differences:

7 - 4 = 3 10 - 7 = 3 13 - 10 = 3

The second differences are constant, so this is a quadratic sequence.

The nth term of a quadratic sequence is given by the formula an^2 + bn + c.

Since the second difference is constant and equal to 3, the coefficient a in the formula is 3/2 = 1.5.

To find the coefficients b and c, we can substitute the first two terms of the sequence into the formula:

For n = 1, the first term is 1, so we have 1.51^2 + b1 + c = 1. This simplifies to 1.5 + b + c = 1.

For n = 2, the second term is 5, so we have 1.52^2 + b2 + c = 5. This simplifies to 6 + 2b + c = 5.

Solving these two equations simultaneously, we find that b = 0.5 and c = -1.

So, the nth term of the sequence is 1.5n^2 + 0.5n - 1.