If the first n terms of positive term sequence {an} and Sn satisfy 10sn = an ^ 2 + 5An + 6, and A1, A3, A15 are equal proportion sequence, then A2010=

If the first n terms of positive term sequence {an} and Sn satisfy 10sn = an ^ 2 + 5An + 6, and A1, A3, A15 are equal proportion sequence, then A2010=

10Sn=(an)²+5an+6
10S(n-1)=(a(n-1))²+5a(n-1）+6
If you subtract the two expressions, you get
5a(n-1)+5an=(an)²-(a(n-1))²
5=an-a(n-1)
So {an} is an arithmetic sequence, the first term A1, tolerance d = 5, so
an=na1+(n-1)n/2
a1*a15=(a3)²
a1*(a1+14d)=(a1+2d)²
5d*a1=2d²
d(5a1-2d)=0
∵d=5
So 5A1 = 2D
a1=2
∴an=a1+(n-1)d=2+(n-1)=5n-3
When n = 2010, we get
a2010=10047