Question

数列a1=1,a2=2.,(an+1+an)/an=(an+2-an+1)/an+1,求an

Answer 1

(an+1+an)/an=(an+2-an+1)/an+1
a(n+1)/an +1=a(n+2)/a(n+1)-1
a(n+2)/a(n+1)-a(n+1)/an=2
{ a(n+1)/an}为等差数列，公差为2
a(n+1)/an=2+（n-1)2=2n
当n≥2时，
a2/a1=2
a3/a2=4
a4/a3=6
a5/a4=8
...............
an/a(n-1)=2(n-1)
an/a1=2×4×6×.....×2(n-1)
=2^(n-1)×(n-1)!
an=(n-1)!2^(n-1)
当n=1时上式成立
an=(n-1)!2^(n-1)（n∈N*)