Question

若数列{Xn}满足:X1=1,X2=3且X(n+1)/Xn=3Xn/Xn-1(n=2,3,4...)

则它的通项公式Xn等于

Answer 1

解
x(n+1)/xn=3xn/x(n-1)
[x(n+1)/xn]/[xn/x(n-1)]=3
x2/x1=3/1=3
数列{xn/x(n-1)}是以3为首项,公比为3的等比数列
∴xn/x(n-1)=(x2/x1)*3^(n-2)=3^(n-1) (n>=2)
又xn=[xn/x(n-1)]*[x(n-1)/x(n-2)]*...*(x2/x1)*x1
=3^(n-1) *3^(n-2) *...*3^1*1
=3^(n-1+n-2...+1)
=3^[n(n-1)/2]
当n=1是也满足上式
∴xn=3^[n(n-1)/2] (n>=1)