a1,a2,...,an為正等比數列,已知S=a1+a2+...+an,T=1/a1+1/a2+...+1/an,試以S,T和n表a1*a2*...*an?

a_1,a_2,...,a_n為正等比數列. 令 r = a_n/a_(n-1), 則 a_k = a_1 × r^(k-1). S = Σ[a_1 × r^(k-1)] = a_1(1-r^n)/(1-r) T = Σ[ (1/a_1) × r^[-(k-1)] = (1/a_1)(1-r^(-n))/(1-r^(-1)) = (1/a_1)r^(-n+1)(1-r^n)/(1-r) S/T = a_1^2 r^(n-1) a_1 × a_2 × … × a_n = a_1^n r^[1+2+...+(n-1)] = a_1^n r^[n(n-1)/2] = (a_1^2 r^(n-1))^(n/2) = (S/T)^(n/2)