Question

n+1 = 2010^2 + 2011^2 find √(2n+1)

Answer 1

$n + 1 = {2010}^{2} + {2011}^{2} \\ n + 1 = {(2011 - 2010)}^{2} + 2 \times 2011 \times 2010 \\ n + 1 = 1 + 2 \times 2011 \times 2010$
This gives that
$n = 2 \times 2011 \times 2010 \\ 2n + 1 = 2 \times 2011 \times 2010 \times 2+ 1 \\ 2n + 1 = 4022 \times 4020 + 1$
Now it is easy to calculate.