Question

已知a1=1,an+1=an+2^n,求an.

Answer 1

a1=1=2^0

a2=a1+2^1=2^0+2^1

a3=a2+2^2=2^0+2^1+2^2

a4=a3+2^3=2^0+2^1+2^2+2^3

∴an=2^0+2^1+2^2+2^3+……+2^(n-1),为一等比数列,根据等比数列公式可知

an=1*(1-2^n)/(1-2)=2^n-1