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

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The discussion centers on whether there is only one arithmetic sequence where the sum of the first n terms equals n^2. The derived formula for the nth term is an = 2n - 1, indicating a unique sequence. The sum of n terms can be expressed as a quadratic polynomial, which must have specific coefficients for the sequence to hold true. The pattern observed shows that the first term is 1, and subsequent terms increase by 2, confirming the uniqueness of the sequence. Thus, there is indeed only one arithmetic sequence that satisfies the condition.
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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?
 
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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.
 
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?
 

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