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a(n) = aq^(n-1),
q=1时,a(n) = a, s(n) = na, 3 = s(9)/s(3) = (9a)/(3a),满足题意,a(9)/a(3) = a/a = 1.
q不为1时,s(n) = a[1-q^n]/(1-q), 3 = s(9)/s(3) = [1-q^9]/[1-q^3].
3 - 3q^3 = 1 - q^9,
0 = q^9 - 3q^3 + 2.
记t = q^3,t不为1.
0 = t^3 - 3t + 2 = t^3 - t^2 + t^2 - 3t + 2 = (t-1)t^2 + (t-1)(t-2) = (t-1)(t^2 + t - 2)=(t-1)(t+2)(t-1),
0 = t+2.
-2 = t = q^3.
a(9)/a(3) = aq^8/(aq^2) = q^6 = (-2)^2 = 4.
综合,有,
公比q=1时,a(9)/a(3) = 1.
公比q不为1时,a(9)/a(3) = 4.
q=1时,a(n) = a, s(n) = na, 3 = s(9)/s(3) = (9a)/(3a),满足题意,a(9)/a(3) = a/a = 1.
q不为1时,s(n) = a[1-q^n]/(1-q), 3 = s(9)/s(3) = [1-q^9]/[1-q^3].
3 - 3q^3 = 1 - q^9,
0 = q^9 - 3q^3 + 2.
记t = q^3,t不为1.
0 = t^3 - 3t + 2 = t^3 - t^2 + t^2 - 3t + 2 = (t-1)t^2 + (t-1)(t-2) = (t-1)(t^2 + t - 2)=(t-1)(t+2)(t-1),
0 = t+2.
-2 = t = q^3.
a(9)/a(3) = aq^8/(aq^2) = q^6 = (-2)^2 = 4.
综合,有,
公比q=1时,a(9)/a(3) = 1.
公比q不为1时,a(9)/a(3) = 4.
1年前
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