已知Un=a^n+a^(n-1)b+a^(n-2)b^2+...+ab^(n-1)+b^n(n∈N*,a>0,b>0),当a=b时,求数列{Un}的前N项和Sn

已知Un=a^n+a^(n-1)b+a^(n-2)b^2+...+ab^(n-1)+b^n(n∈N*,a>0,b>0),当a=b时,求数列{Un}的前N项和Sn

Un=a^n+a^(n-1)b+a^(n-2)b^2+...+ab^(n-1)+b^n
a=b
Un=na^n
consider
1+x+x^2+..+x^n= (x^(n+1)-1)/(x-1)
1+2x+..+nx^(n-1)
=[(x^(n+1)-1)/(x-1)]'
= [nx^(n+1)-(n+1)x^n+1]/(x-1)^2
multiply both side by x
x+2x^2+..+nx^n = x[nx^(n+1)-(n+1)x^n+1]/(x-1)^2
Un = na^n
Sn = a[na^(n+1) -(n+1)a^n+1]/(a-1)^2