Question

x and x + y are the square of two consecutive
natural number. What is the square of the next
natural number?
​

Answer 1

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.

Answer 2

Answer:

Solution

Let the square be

$\sf \ell - 1$

$\ell$

$\ell \: + 1$

Then

Squares are

${ \ell}^{2} - 1 = x$

${ \ell}^{2} = x + y$

x + y - x

(x + y) - x

${ \ell}^{2} - ( { \ell - 1)}^{2}$

x + y - x = l² + l² - 1

y = 2l - 1

Therefore,

Square = x + 2y + 2