SOLUTION: the sum of n terms of an arithmetic series is:Sn=5n^2-11. Determine common difference

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Question 1070612: the sum of n terms of an arithmetic series is:Sn=5n^2-11. Determine common difference
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There is a typo there.
S%5Bn%5D=5n%5E2-11 cannot be the sum of n terms of an arithmetic series,
but S%5Bn%5D=5n%5E2-11n could be .
You could use either expression to find the common difference d=10 ,
but only a sum of the form an%5E2%2Bbn could work to find an actual
arithmetic series, with a first term a%5B1%5D independent of n .

A WAY TO SOLVE IT WITH S%5Bn%5D=5n%5E2-11n :
The sum of n-1 terms is
S%5Bn-1%5D=5%28n-1%29%5E2-11%28n-1%29
S%5Bn-1%5D=5%28n%5E2-2n%2B1%29-11n%2B11
S%5Bn-1%5D=5n%5E2-10n%2B5-11n%2B11
S%5Bn-1%5D=5n%5E2-21n%2B16
and since S%5Bn%5D=S%5Bn-1%5D%2Ba%5Bn%5D where a%5Bn%5D is term number n ,
we can find an expression for a%5Bn%5D=S%5Bn%5D-S%5Bn-1%5D .
a%5Bn%5D=5n%5E2-11n-%285n%5E2-21n%2B16%29
a%5Bn%5D=5n%5E2-11n-5n%5E2%2B21n-16%29
a%5Bn%5D=10n-16%29
a%5Bn%5D=10n-10-6%29
a%5Bn%5D=10%28n-1%29-6%29
That gives us highlight%28d=10%29 as the common difference,
and a%5B1%5D=-6 as the first term.
We know that in an arithmetic sequence
with a first term a%5B1%5D ,
and a common difference d ,
term number n is
a%5Bn%5D=a%5B1%5D%2Bd%28n-1%29 ,
so d is the coefficient of %28n-1%29 ,
and the other term is a%5B1%5D .

WHY WE KNOW S%5Bn%5D=5n%5E2-11 IS WRONG:
We have the "formula" S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29n%2F2
In that expression n is a factor.
The expression is a quadratic (degree 2) polynomial with
S%5Bn%5D=%28%282a%5B1%5D-d%29%2Bnd%29n%2F2
S%5Bn%5D=%28dn%5E2%2B%282a%5B1%5D-d%29n%29%2F2
S%5Bn%5D=%28d%2F2%29n%5E2%2B%28%282a%5B1%5D-d%29%2F2%29n