已知正项等比数列an满足:a3=a2+2a1,若存在两项am,an使得根号下aman=4a1,1/m+4/n的最小值为

Question

已知正项等比数列an满足:a3=a2+2a1,若存在两项am,an使得根号下aman=4a1,1/m+4/n的最小值为

Answer 1

a(n) = aq^(n-1),a>0,q>0.
a(3)=aq^2 = a(2)+2a(1) = aq+2a,
q^2 = q + 2,
0 = q^2 - q - 2 = (q-2)(q+1),q=2.
a(n) = a*2^(n-1).
a(m)=a*2^(m-1).
4a(1) = 4a = [a(m)a(n)]^(1/2) = a*2^[(m+n-2)/2],
4 = 2^[(m+n-2)/2] = 2^2,
2 = (m+n-2)/2,
4 = m+n-2,
m=6-n.
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