Find the pattern in the sequence 2, 6, 12, 20 and determine the 34th term

Sure, let's find the pattern in the sequence first.

The sequence is: 2, 6, 12, 20

If we look at the differences between the terms:

6 - 2 = 4 12 - 6 = 6 20 - 12 = 8

We can see that the differences are increasing by 2 each time. This suggests that the sequence is quadratic, and the nth term can be represented as an^2 + bn + c.

To find the coefficients a, b, and c, we can set up a system of equations using the first three terms of the sequence (where n is the term number):

For n = 1, the term is 2, so a1 + b1 + c = 2 For n = 2, the term is 6, so a4 + b2 + c = 6 For n = 3, the term is 12, so a9 + b3 + c = 12

Solving this system of equations, we find that a = 2, b = -1, and c = 1. So the nth term of the sequence is 2n^2 - n + 1.

To find the 34th term, we substitute n = 34 into this formula:

2*(34)^2 - 34 + 1 = 2278.