Use differences to find a pattern in the sequence. 6,13,30,65,126,221,358

To find a pattern in the sequence, we will calculate the differences between consecutive terms:

Second differences:
13-6 = 7
30-13 = 17
65-30 = 35
126-65 = 61
221-126 = 95
358-221 = 137

Third differences:
17-7 = 10
35-17 = 18
61-35 = 26
95-61 = 34
137-95 = 42

We notice that the third differences are all increasing by 8. This indicates that the sequence may be described by a cubic equation of the form an^3 + bn^2 + cn + d .

We can then proceed to find the specific cubic equation that fits the given sequence.

Let's assume the cubic equation to be an^3 + bn^2 + cn + d , where n is the position in the sequence starting from 1.

Plugging in n=1,2,3,4,5,6,7 for the sequence terms 6,13,30,65,126,221,358 , we get the following set of equations:

a + b + c + d = 6
8a + 4b + 2c + d = 13
27a + 9b + 3c + d = 30
64a + 16b + 4c + d = 65
125a + 25b + 5c + d = 126
216a + 36b + 6c + d = 221
343a + 49b + 7c + d = 358

Solving the system of equations, we find:
a=\frac{4}{3},\quad b=-3,\quad c=\frac{20}{3},\quad d=1

Therefore, the cubic equation that describes the given sequence is:
f(n)=\frac{4}{3}n^3-3n^2+\frac{20}{3}n+1