Question

在数列an中，a1等于1，an加1等于2an加2的n次方，设bn等于an除以2的n减一次方，证明bn是等差数列

Answer 1

a1=1
a(n+1) = 2an + 2^n
a(n+1)/2^(n+1) -an/2^n = 1/2
=> {an/2^n} 是等差数列, d=1/2
an/2^n - a1/2^1 = (1/2)(n-1)
an/2^n - 1/2 = (1/2)(n-1)
an = n.2^(n-1)
bn = an/2^(n-1) = n
=> {bn} 是等差数列, b1=1, d=1

Answer 2

a(n+1)/2^n-an/2^(n-1)=1b(n+1)-bn=1

Answer 3

a(n+1)=2an+2^n,bn=an/2^(n-1),b(n+1)=a(n+1)/2^n,b1=a1/2^0=1
a(n+1)/2^n=an/2^(n-1)+1,b(n+1)=bn+1,bn为首项为1公差为1的等比数列，bn=n