Question

在数列｛an｝中，a1=2,an+1=an+ln(1+1/n),则an=?

求过程

Answer 1

an+1=an+ln(1+1/n)=an+ln((n+1)/n)=an+ln(n+1)-lnn
则有a2=a1+ln2-ln1
a3=a2+ln3-ln2
a4=a3+ln4-ln3
.....
an=an-1+lnn-ln(n-1)
等式两边相加得 化简后
an=a1+lnn-ln1=a1+lnn=2+lnn