X and x + y are the square of two consecutivenatural number. What is t

1.	X and x + y are the square of two consecutivenatural number. What is the square of the nextnatural number?
Answer» Question. If $x,x+y$ are the square of two consecutive NATURAL NUMBERS, what is the next square? Keys. For convenience, let the consecutive numbers be $a-1,a,a+1$ . We are going to look for the DIFFERENCE. Solution. Given that, two squares are $x,x+y$ . What is the next square? Let the three consecutive numbers be $a-1,a,a+1$ . The difference of the 1st and 2nd squares is $(x+y)-x=a^2-(a-1)^2$ . $\therefore y=2a-1$ The difference of the 2nd and 3rd squares is $(a+1)^2-a^2=2a+1=y+2$ . So, the next square is $\boxed{x+2y+2}$ . Verification We can find the next square without actual calculation. Given that two squares are $99^2=9801, 100^2=10000$ the next square is $9801+2\times 199+2=10201$ . As $101^2=10201$ , we FOUND a new square using this LOGIC.