Question

有一个公比为2的等比数列,共10项,且以2为底各项的对数的和为25,试求原数列各项的和.

Answer 1

按题意可知

Sn=a1(2^10-1)

由对数函数性质

log(2)a1+log(2)a2+……+log(2)a10

=log(2)a1a2……a10

=25

a1a2……a10=2^25

且

a2=a1q=2a1

a3=a1q^2=(2^2)a1

则

a1a2……a10

=(a1^10)(2^45)=2^25

则

a1^10=2^(-20)

a1=1/4

则

Sn=a1(2^10-1)

=(2^10-1)/4

=2^8-1/4