｛an}等比数列,且an>0,a1*a100=100 则 lga1+lga2+...+lga100=

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｛an}等比数列,且an>0,a1*a100=100 则 lga1+lga2+...+lga100=
｛an}等比数列,且an>0,a1*a100=100 则 lga1+lga2+...+lga100=

｛an}等比数列,且an>0,a1*a100=100 则 lga1+lga2+...+lga100=
lga1+lga2+...+lga100=lga1*a2*.*a99*a100
由｛an}等比数列,得
a1*a100=a2*a99=a3*a98=.=a50*a51=100
而a1*a2*.*a99*a100=a1*a100*a2*a99*a3*a98*.*a50*a51=100^(100/2)=100^50=10^100
从而 lga1+lga2+...+lga100=lga1*a2*.*a99*a100
=lga1*a100*a2*a99*a3*a98*.*a50*a51
=lg10^100
=100

100

a1*a100=a1*a1*q^99=(a1^2)*(q^99)
a2*a99=a1q*a1q^98=(a1^2)*(q^99)
a3*a98=a1q^2*a1q^97=(a1^2)*(q^99)
..
a49*a52=a1q^48* a1q^51=(a1^2)*(q^99)
a50*a51=a1q^49* a1q^50=(a1^2)*(q^99)
lga1+lga2+..+lga100=50lg(a1*a100)=50*lg100=100

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