Question

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

Answer 1

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

Answer 2

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

Answer 3

100