Calculus 2 Geometric

Question

Consider the sequence 2,7,15,26,40,57.... with a0=2 

By looking at the differences between terms, express the sequence as a sequence of partial sums, bn

Then find a closed formula for the sequence by computing the nth partial sum. 

Answer 1

The difference table will show that the second differences are constant and equal 3.

This means the function (call it u_n) is a quadratic.

The first difference is 3n+C₁ and u_n=(3/2)(n²-n)+2n.

This solution comes about by knowing about calculus of finite differences which I can explain, but not in answer to this question.

You can also get to this answer using the Gregory-Newton formula (which you can look up).