= & gta(n+ 1)^2=2an^2+ 1

Let bn = an 2,

Then b 1 = 1, b (n+ 1) = 2bn+ 1.

b(n+ 1)+ 1 = 2(bn+ 1)

So bn+ 1 is a geometric series with a common ratio of 2, and b 1+ 1=2.

bn+ 1=2^n

bn=2^n- 1

an=√bn=√2^n- 1