MHB Is the Sum of n^2 Terms in an Arithmetic Sequence Limited to 1?

[stamenkovoca02]

How many different arithmetic sequences have the sum of the first n terms n^2?
solution an= 2n-1.Does that mean there is only one arithmetic sequence?

 

[Evgeny.Makarov]

If the sum of the first $n$ terms must equal $n^2$ for all $n$, then yes, such sequence is unique. To see why, write the sum of $n$ terms using the first term $a_1$ and the difference $d$. This is going to be a quadratic polynomial. Its leading coefficient has to be equal to 1, and the other two have to be 0.

 

[HOI]

The first term must be 1. The second term must satisfy 1+ x= 4 so x= 3. The third term must satisfy 4+ x= 9 so x=5. The fourth term must satisfy 9+ x= 16 so x= 7.. The fifth term must satisfy 16+ X= 25 so x=9. Do you see a pattern?