∑ i=1 to n, √[1+(1/i^2)+(1/(1+i^2))] = n(n+2)/n+1

First I did the base case of p(1) showing 3/2 on the LHS equals the 3/2 on the RHS.

Then I assumed p(k) and wrote out the formula with k in it.

Then prove p(k+1)= p(k)+ √[1+1/(k+1)^2+1/(k+2)^2]

=k(k+2)/k+1 + √[1+1/(k+1)^2+1/(k+2)^2]

Then I squared each to get rid of the square root.

(k(k+2)/(k+1))^2+ (k+1)^2/(k+1)^2 + 1/(k+1)^2 + 1/(k+2)^2

Now I'm stuck any Guidance would be great thanks!