Menu Close

Is-there-a-solution-to-i-a-n-i-i-a-n-i-n-a-Z-




Question Number 3193 by Filup last updated on 07/Dec/15
Is there a solution to:  Σ_(i=a) ^n i=Π_(i=a) ^n i                (n, a)∈Z
Isthereasolutionto:ni=ai=ni=ai(n,a)Z
Commented by Filup last updated on 07/Dec/15
Σ_(i=a) ^n i=(1/2)n(a+n)  Π_(i=a) ^n i=((n!)/((a−1)!))    ∴(1/2)n(a+n)=((n!)/((a−1)!))
ni=ai=12n(a+n)ni=ai=n!(a1)!12n(a+n)=n!(a1)!
Commented by prakash jain last updated on 07/Dec/15
n=a   n=a Π_(i=a) ^n i=a=Σ_(i=a) ^n i  your formula for Σ_(i=a) ^n i is incorrect. it should  be=((n−a+1)/2)(a+n)  If n>a and a>1  a(a+1)≠a+a+1  so no solutions for n>a  n<a LHS=0=RHS
n=an=ani=ai=a=ni=aiyourformulaforni=aiisincorrect.itshouldbe=na+12(a+n)Ifn>aanda>1a(a+1)a+a+1sonosolutionsforn>an<aLHS=0=RHS
Commented by Filup last updated on 07/Dec/15
Thanks for the correction!  What if     a,n∈R?  Same result?
Thanksforthecorrection!Whatifa,nR?Sameresult?
Commented by Filup last updated on 07/Dec/15
What if  a+(a+1)+...+(n)=a(a+1)...n  for  (a, n)∈R
Whatifa+(a+1)++(n)=a(a+1)nfor(a,n)R
Commented by prakash jain last updated on 07/Dec/15
If a and n are real. Then you would get  a equation of degree n−a+1 with n−a+1  solutions. Some of them will be complex.
Ifaandnarereal.Thenyouwouldgetaequationofdegreena+1withna+1solutions.Someofthemwillbecomplex.

Leave a Reply

Your email address will not be published. Required fields are marked *