Let TN = 1 / A1A2 + 1 / a2a3 + 1 / a3a4 + +1 / Anan + 1, find TN

Let TN = 1 / A1A2 + 1 / a2a3 + 1 / a3a4 + +1 / Anan + 1, find TN

{an} is an arithmetic sequence,
a1=S1=1,S2=(1/2)*4+(1/2)*2=3
a2=S2-S1=2
d=1
an=1+(n-1)*1=n
1/[an*a(n+1)]=1/[n(n+1)]=1/n-1/(n+1)
Tn=1-1/2+1/2-1/3+1/3-1/4+.+1/n-1/(n+1)
=1-1/(n+1)
=n/(n+1)