Question

数列{an}中,a1=1,an=an-1+2^n,求an

Answer 1

an=a(n-1)+2^n,
an-a(n-1)=2^n

an-a(n-1)=2^n
a(n-1)-a(n-2)=2^(n-1)
..........
a3-a2=2^3
a2-a1=2^2
以上等式相加得
an-a1=2^2+2^3+....+2^n
an-a1=4*[1-2^(n-1)]/(1-2)
an-a1=4*[2^(n-1)-1]
an-1=2^(n+1)-4
an=2^(n+1)-3

Answer 2

an=an-1+2^n
an/2^n=(an-1)2^n-1+1
an/2^n-2=1/2*[(an-1)/2^(n-1)-2]
[an/2^n-2]/[(an-1)2^(n-1)-2]=1/2 等比公比1/2首项-3/2
an/2^n-2=-3/2*1/2^(n-1)=-3/2^n
an=[-3/2^n+2]*2^n=2^(n+1)-3