Question

已知数列{an}满足an+1/an=n+2/n(n∈N*)且a1=1,则an=

Answer 1

因为a(n+1)/an=(n+2)/n
所以有an/a(n-1)=(n+1)/(n-1)
a(n-1)/a(n-2)=n/(n-2)
a(n-2)/a(n-3)=(n-1)/(n-3)
......
a4/a3=5/3
a3/a2=4/2
a2/a1=3/1
有叠乘知：an/a1=n(n+1)/2，再将a1带入就可以得到：an=n(n+1)/2

Answer 2

a(n+1)/an=(n+2)/n

an/a(n-1)=(n+1)/(n-1)
a(n-1)/a(n-2)=n/(n-2)
a(n-2)/a(n-3)=(n-1)/(n-3)
......
a4/a3=5/3
a3/a2=4/2
a2/a1=3/1
各式相乘
an/a1=n(n+1)/2
an=n^2/2+n/2