等比数列{an},a1+a2+a3+a4+a5=8,1/(a1)+1/(a2)+1/(a3)+1/(a4)+1/(a5)=2,求a3.(a后的数字都为下标,是序号)
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等比数列{an},a1+a2+a3+a4+a5=8,1/(a1)+1/(a2)+1/(a3)+1/(a4)+1/(a5)=2,求a3.(a后的数字都为下标,是序号)
等比数列{an},a1+a2+a3+a4+a5=8,1/(a1)+1/(a2)+1/(a3)+1/(a4)+1/(a5)=2,求a3.(a后的数字都为下标,是序号)
等比数列{an},a1+a2+a3+a4+a5=8,1/(a1)+1/(a2)+1/(a3)+1/(a4)+1/(a5)=2,求a3.(a后的数字都为下标,是序号)
a1+a2+a3+a4+a5=8.A
a1+a2+a4+a5=8-a3.B
a3^2=a1a5=a2a4.C
1/(a1)+1/(a2)+1/(a3)+1/(a4)+1/(a5)=2
a3^3(a1+a2+a4+a5)/a1a2a3a4a5=2.D
将B,C代入D得
a3^3(8-a3)/a3^5=2
8-a3=2a3^2
2a3^2+a3-8=0
a3=[-1±√65]/4
a(n) = aq^(n-1), n = 1,2,...
1/a(n) = 1/[aq^(n-1)] = (1/a)(1/q)^(n-1), n = 1,2,...
若q = 1,则，8 = a(1) + a(2) + ... + a(5) = 5a, a = 8/5.
2 = 1/a(1) + 1/a(2) + ... + 1/a(5) = 5/a = 5/(8/5) =...
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a(n) = aq^(n-1), n = 1,2,...
1/a(n) = 1/[aq^(n-1)] = (1/a)(1/q)^(n-1), n = 1,2,...
若q = 1,则，8 = a(1) + a(2) + ... + a(5) = 5a, a = 8/5.
2 = 1/a(1) + 1/a(2) + ... + 1/a(5) = 5/a = 5/(8/5) = 25/8,矛盾。
因此，q不等于1.
8 = a(1) + a(2) + ... + a(5) = a[q^5 - 1]/[q-1],
2 = 1/a(1) + 1/a(2) + ... + 1/a(5) = (1/a)[(1/q)^5 - 1]/[1/q - 1] = 1/(aq^4)[q^5 - 1]/[q-1] = 8/(a^2q^4)
a^2q^4 = 4,
a(3) = aq^2 = 2或-2.
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