设n为自然数,在数1/n和n+1之间插入n个正数,使n+2个正数成等比数列

Question

设n为自然数,在数1/n和n+1之间插入n个正数,使n+2个正数成等比数列

Answer 1

设公比为q
a1 = 1/n
a(n+2) = n+1 = 1/n * q^(n+1)
q = [n(n+1)] ^ (1/(n+1))
a1 = 1/n
a2 = (1/n) * [n(n+1)] ^ (1/(n+1))
a3 = (1/n) * [n(n+1)] ^ (2/(n+1))
.
a(n+2) = (1/n) * [n(n+1)] ^ ((n+1)/(n+1)) =n+1