等比数列{an}的首项为一公比为q【q不等于一】,前n项和为Sn,则数列{1／an}的前n项和是

题目

等比数列{an}的首项为一公比为q【q不等于一】,前n项和为Sn,则数列{1／an}的前n项和是

答案

数列{1／an}的前n项和Tn
=1/a1+1/a2+1/a3+……+1/an
=1/a1+1/a1q+1/a1q^2+……+1/a1q^(n-1)
=1/a1*[1-(1/q)^n]/(1-1/q)
=(q^n-1)/[a1(q-1)*q^(n-1)]