已知a,b,c依次成等差数列,它们的和为33,又lg(a-1),lg(b-5),lg(c-6)也成等差数列,求a,b,c.

a,b,c依次成等差数列
2b=a+c 1
a+b+c=33 2
lg(a-1),lg(b-5),lg(c-6)
2lg(b-5)=lg(a-1)+lg(c-6)
lg(b-5)^2=lg(a-1)(c-6)
(b-5)^2=(a-1)(c-6) 3
1式代入2式得
3b=33
b=11
a+c=22 -->a=22-c 4
(11-5)^2=(a-1)(c-6)=36 5
4式代入5式得
(22-c-1)(c-6)=36
(21-c)(c-6)=36
21c-126-c^2+6c=36
c^2-27c+162=0
(c-18)(c-9)=0
c=18或 c=9
a=4或 a=13
所以a,b,c是13,11,9或4,11,18