You are absolutely right. The series given is not an AP or a GP as neither is the difference between consecutive terms common nor is the ratio of consecutive terms the same. But if you notice carefully the series is just made up by the squares of consecutive numbers. 1= 1^2 , 4= 2^2 , 9= 3^3, 16= 4^2 , 25= 5^2 and so on.
Now the relation for the sum of the first n squares is given by the relation: n*(n+1)*(2n+1) / 6
Therefore the sum of the first n terms of the series is
= n*(n+1)*(2n+1) / 6.
To find the sum 1+4+9+16++25+36+....
This could be written in the form;
Sn = 1+2^2+3^2+....n^2 which is equal to n(n+1)(2n+1)/6.