数列{an}中,a1=1,an+1=2an+2的n+1次方,证bn=an/2的n次方等比并求an.

数列{an}中,a1=1,an+1=2an+2的n+1次方,证bn=an/2的n次方等比并求an.
数列{an}中,a1=1,an+1=2an+2的n+1次方,证bn=an/2的n次方等比
并求an.

数列{an}中,a1=1,an+1=2an+2的n+1次方,证bn=an/2的n次方等比并求an.
(1)
a(n+1)=2an+2^(n+1)
等式两边同除以2^(n+1)
a(n+1)/2^(n+1)=an/2ⁿ+1
a(n+1)/2^(n+1)-an/2ⁿ=1,为定值
a1/2=1/2,数列{an/2ⁿ}是以1/2为首项,1为公差的等差数列
bn=an/2ⁿ,数列{bn}是以1/2为首项,1为公差的等差数列(不是等比,是等差)
(2)
an/2ⁿ=1/2+1×(n-1)= n-1/2=(2n-1)/2
an=[(2n-1)/2]·2ⁿ=(2n-1)·2^(n-1)
n=1时,a1=(2×1-1)×1=1,同样满足通项公式
数列{an}的通项公式为an=(2n-1)·2^(n-1)