What is the pattern rule for this pattern 3-4-8-17-33?

Answer 1

Starting with 3, the number added to each term to get the next term is the square of each positive integer in turn, starting with 1:

3 + 1² = 4

4 + 2² =. 8

8 + 3² = 17

17 + 4² = 33

33 + 5² = 58

...and so on.

One can discover this by looking at the sequence of successive differences between the original given terms:

4-3, 8-4, 17-8, 33-17, ... or

1, 4, 9, 16, ... which is

1², 2², 3², 4², ...

The recursive formula is

Σ(1) = 3; Σ(n+1) = Σ(n) + (n-1)²

Answer 2

After the initial number 3, the next number is generated by adding the square of numbers increasing by one at each step and starting with 12.

3 + 12 = 4

4 + 22 = 8

8 + 32 = 17

17 + 42 = 33....and so on

The series formula is U(n) = n3/3 - n2/2 + n/6 + 3, which can also be written as :-

3 + n/6(2n2 - 3n + 1) = 3 +n/6(2n - 1)(n - 1)