Question

已知等比数列{an}的首项a1=2,公比q=3,Sn是它的前n项和.求证：Sn+1/Sn

Answer 1

Sn=2(1-3^n)/(1-3)=3^n-1

S(n+1)=3*3^n-1

S(n+1)/Sn=(3*3^n-1)/(3^n-1)=(3*3^n-3+2)/(3^n-1)=3+2/(3^n-1)

(3n+1)/n =3+1/n

证明原式只需证明2/(3^n-1)≤1/n

即证明(3^n-1)≥2n

设y=3^n-2n-1

当n=1时,y=0,

当n从1增大时,3^n不2n增加的快,所以y≥0

即3^n-2n-1≥0,

3^n-1>2n

原式得证