a1＞1,公比q＞0,设bn=log2底an,且b1+b3+b5=6,b1b3b5=0,证明bn为等差数列

Question

a1＞1,公比q＞0,设bn=log2底an,且b1+b3+b5=6,b1b3b5=0,证明bn为等差数列

Answer 1

证：
an=a1q^(n-1)
bn=log2(an)=log2[a1q^(n-1)]=log2(a1)+(n-1)log2(q)
b1=log2(a1)
bn-b(n-1)=log2(a1)+(n-1)log2(q)-log2(a1)-(n-2)log2(q)=log2(q),为定值.
数列{bn}是以log2(a1)为首项,log(q)为公差的等差数列.
证明其实到这里就可以了,后面的两个等式是多余的,不过利用两个等式可以求出{bn}具体的数值表达形式.
b1+b3+b5=b3-2d+b3+b3+2d=3b3=6 b3=2
b1b3b5=0 (b3-2d)b3(b3+2d)=0
2(2-2d)(2+2d)=0
1-d²=0 d=1或d=-1(an是对数的真数,an>0,d=-1舍去)
b1=b3-2d=2-2=0
{bn}是以0为首项,1为公差的等差数列.