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

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SUMMARY

The discussion confirms that there is a unique arithmetic sequence where the sum of the first n terms equals n^2. The solution is derived from the formula an = 2n - 1, indicating that the first term must be 1 and subsequent terms follow a specific pattern. The terms are generated by adding consecutive odd integers, demonstrating that the sequence is uniquely defined for all n. This conclusion is supported by the quadratic nature of the sum of the terms.

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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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