Algebra problem (45)

Problem

This problem was posted by Lahcene Belhadj in Facebook group "Classical Mathematics".
The solution is my own.

Calculate:



Solution





let:



calculate Sn for n = 1, 2 3, 4, .....



and we see



which is certainly correct for n=1 to n=6 but further?
To obtain general integrity we need to prove that Sn+1 = Sn + tn+1.
Because if Sn is true for n=1 than also for n=2,3,....



It works!

What happens to Sn if n grows very large?



so K = 22 = 4.