Question

An=2^n*sin(π/3n)比较An+1（n+1是下标）和An的大小

Answer 1

An=2^n*sin(π/3n)

An+1=2^(n+1)*sin(π/3(n+1))

An+1/An = 2 sin(π/3(n+1))/sin(π/3n)

因为

π/3n 0

2π/3(n+1) 2sin(π/3(n+1))cos(π/3(n+1))

= sin(2π/3(n+1))

= sin(π/3n * 2n/(n+1))

= sin(π/3n * (n+n)/(n+1))

>= sin(π/3n * (n+1)/(n+1))

= sin(π/3n)

所以 An+1/An = 2 sin(π/3(n+1))/sin(π/3n) > sin(π/3n)/sin(π/3n) =1

An+1 >An