Question

There are _________ natural numbers between $n^2$ and ${(n+1)}^2$.

Answer 1

Explanation:

Lets take any two consecutive natural numbers $n=2$ and $n+1=2+1=3$.

$\therefore n^2=2^2=2\times 2=4$ and

$\therefore {\left(n+1\right)}^2={\left(3\right)}^2=3\times 3=9$

Therefore, Natural numbers between the squares of $2$ and $3$ are $5,6,7,8$ which are $2\times n=2\times 2=4$ numbers.

Lets take any other two consecutive natural numbers $n=5$ and $n+1=5+1=6$.

$\therefore n^2=5^2=5\times 5=25$ and

$\therefore {\left(n+1\right)}^2={\left(6\right)}^2=6\times 6=36$

Therefore, Natural numbers between the squares of $5$ and $6$ are $26,27,28,29,30,31,32,33,34,35$ which are $2\times n=2\times 5=10$ numbers.

Therefore, there are $2n$ natural numbers between $n^2$ and ${(n+1)}^2$.