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

AI Thread Summary
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.
stamenkovoca02
Messages
4
Reaction score
0
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?
 
Mathematics news on Phys.org
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?
 

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Similar threads

MHB Arithmetic Progression: Expressing d in Terms of x,y,z,n
Replies
1
Views
1K
MHB Determining if a sequence is arithmetic
Replies
5
Views
2K
MHB Arithmetic Sequence Confusion // a{n-1} and a{n+1}
Replies
1
Views
2K
MHB Sum of First 20 Terms of Arithmetic Progression with Even Terms Removed
Replies
7
Views
2K
B Looking for context for a textbook quote
Replies
9
Views
2K
MHB What is the Sum of an Arithmetic Sequence?
Replies
8
Views
2K
A Limit of ##i^\frac{1}{n}## as ##n \to \infty##
Replies
14
Views
2K
I Is e^pi a unique transcendental, or perhaps not transcendental?
Replies
1
Views
2K
B Indexing Sequences: Do We Start at 0 or 1?
Replies
3
Views
3K
MHB Number patterns and sequences - Tn Term
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top