Question

用数学归纳法证明：1/4+1/4^2+…+1/4^n=1/3-1/(3●4^n)

Answer 1

1).当n=1时，1/4=1/3-1/(3×4),
2).设n=k时原式成立，即1/4+1/4^2+…+1/4^k=1/3-1/(3●4^k)
则1/4+1/4^2+…+1/4^k+1/4^(k+1)
=1/3-1/(3●4^k)+1/4^(k+1)
=1/3-4/[3●4^(k+1)]+3/[4^(k+1)*3]
=1/3-1/[3●4^(k+1)
综合1).、2).得1/4+1/4^2+…+1/4^n=1/3-1/(3●4^n)

Answer 2

由Sn+1=Sn+1/4^n+1
=1/4+1/4^2+…+1/4^n+1/4^n+1
=1/3-1/(3●4^n)+1/4^n+1
=1/3-4/(3●4^n+1)+1/4^n+1
=1/3-(1/4^n+1(4/3-1))=1/3-1/(3●4^n)，所以1/4+1/4^2+…+1/4^n=1/3-1/(3●4^n)在第n+1项仍然成立，证明完毕