Question

Which statement is true about consecutive natural numbers?
(A) The numbers between the difference of consecutive numbers is $2n + 1$
(B) The non-perfect square numbers between the square of consecutive numbers is $2n$
(C) The sum of the squares of two consecutive numbers is never a perfect square.
(D) ${n^2} - 1$ is the standard form of the difference between two consecutive numbers.

Answer 1

Hint: We are given some statements and we have to find which statement is true among them. The statement
Is about consecutive natural numbers. By using the following concept about natural numbers we can verify whether it is true or false. By applying the concept that the number between the difference of consecutive numbers is $2n - 1$. The second concept used is the non-perfect square number between the square of consecutive numbers is $2n$. The third concept used is the sum of the squares of two consecutive numbers that can be a perfect square. And the fourth concept used is the standard form of difference between two consecutive numbers is $1$. By applying these concepts we can find the correct statement.

Complete step-by-step answer:
Step1: We are given the first statement as the numbers between the difference of consecutive numbers is $2n + 1$. If ${n^2}$ is the greater of the two consecutive squares, then the difference will be $2n - 1$. So for example, you have two consecutive squares with the values of $25$ and $36$, their difference will be $11$. Notice that $11 = 2 \times 6 - 1$, so it confirms to $2n - 1$. All perfect squares are also the total sum of all consecutive integers up to $2n - 1$.
${1^2} = 1$
${2^2} = 1 + 3$
${3^2} = 1 + 3 + 5$
${4^2} = 1 + 3 + 5 + 7$
Notice also that when summing all consecutive odd number, $2n - 1$
Hence, the given statement is not true.
Step2: We are given the non-perfect square numbers between the squares of consecutive numbers $2n$. Both ${n^2}$ and ${\left( {n + 1} \right)^2}$ are perfect square numbers and they are consecutive perfect squares.
All the numbers between them are non-perfect squares.
$ = {\left( {n + 1} \right)^2} - {n^2} - 1$
Using the formula (a+b)2 =a2+b2+2ab
$ = {n^2} + 2n + 1 - {n^2} - 1$
On solving
$ = 2n$
There are $2n$ non-perfect square numbers. Hence the given statement is true.
Step3: We are given the next statement that the sum of the squares of two consecutive numbers is never a perfect square. But the sum of the squares of two consecutive numbers can be a perfect square also. It is not possible that there will never be a perfect square. It depends on the numbers only. Hence the statement is not true.
Step4: We are given the statement ${n^2} - 1$ is the standard form of the difference between two consecutive numbers. Natural numbers are counting numbers or positive integers. The set of natural numbers is denoted by $N$.
So $N$$ = \{ 1,2,3,4,5,6,7,8,9,10,11.....\} $
Two consecutive natural numbers are those which are next to each other. i.e , $2,3$ or $6,7$ or $9,10$ and so on. The difference between them is $2 - 1,7 - 1,10 - 9,$ which are all the same namely $1$.
Hence the statement is not true.

Hence , the option (B) is correct.

Note: In case of consecutive numbers use the concept and take the single digit numbers to explain the concept by using the single digit numbers we can easily understand the concept and don’t get confused always use the numbers to understand these concepts.