a(n+1)=(2/an+3)/3/an=(2+3an)/3=2/3+an
a(n+1)-an=2/3
等差数列
an=a1+2/3 *(n-1)=2n/3+1/3
a1a2=1*5/3=3/3*5/3=3*5/9
a2a3=5/3*7/3=5*7/9
a1a2-a2a3=4*5/9=4*(4+1)/9 (1)
a3a4=7/3*9/3=7*9/9
a4a5=9/3*11/3=9*11/9
a3a4-a4a5=4*9/9=4*(4*2+1)/9 (2)
...
a2n-1a2n-a2na2n+1=4*(4*n+1)/9 (n)
上述n个式子相加：
Tn=(4/9)(4+4*2+4*3+...+4*n+n)=(4/9)(4*n(n+1)/2+n)
=(4/9)(2n^2+3n)