Question

数列{an}中 a1,a3,a4三项成等差数列。 则如果{an}是等比数列，则公比q为多少？

对了，q不等于1

Answer 1

{an}为等比数列，则a3 = a1q^2，a4 = a1q^3，且a1≠0
a1，a3，a4为等差数列，即a1+a4 = 2a3，a1 + a1q^3 = 2a1q^2
两边约去a1得1+q^3 = 2q^2
q^3 - 2q^2 + 1 = 0
q^3 - q^2 - (q^2 -1) = 0
q^2(q-1) - (q+1)(q-1) = 0
(q^2-q-1) (q-1) = 0
q^2 - q -1 = 0或q-1=0
即q = 1/2 ± 根号5 / 2或者q = 1

Answer 2

设公比为q 则
a3=a1*q^2
a4=a1*q^3
又因为a1,a3,a4三项成等差数列
则 a3-a1=a4-a3
则 q=1或者q=【(根号5)+1】/2或者 q=【-(根号5)+1】/2

Answer 3

a3 = a1q^2，a4 = a1q^3，且a1≠0
a1，a3，a4为等差数列，即a1+a4 = 2a3，a1 + a1q^3 = 2a1q^2
q^3 - 2q^2 + 1 = 0
q^3 - q^2 - (q^2 -1) = 0
q^2(q-1) - (q+1)(q-1) = 0
q^2 - q -1 = 0或q-1=0
即q = 1/2 ± 根号5 / 2或者q = 1
同时 q不等于1，那么q=q = 1/2 ± 根号5 / 2

Answer 4

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