已知数列an,满足an=9^n(n1)/10^n,试问an有没有最大项?若有求出最大值.是9的n次方再乘以.

Question

已知数列an,满足an=9^n (n 1)/10^n,试问an有没有最大项?若有求出最大值.是9的n次方再乘以.

Answer 1

an=9^n*[(n+1)]/10^n
=(9/10)^n*[(n+1)]
则: a(n+1)/an
={(9/10)^(n+1)*[(n+2)]}/{(9/10)^n*[(n+1)]}
=(9/10)*[(n+2)/(n+1)]
=(9/10)*[1+1/(n+1)]
令a(n+1)/an≥1,得1≤n≤8,
令a(n+1)/an≤1,得n≥8,
故当n=8时,a8=a9
n≤7时,ana(n+1) 数列递减
所以最大值为a8和a9
a8=a9=9^9/10^8.