If you want to recall what are perfect square numbers then please refer to my article on What are the perfect square numbers. This article asks a question that how many natural numbers lie between two consecutive perfect square numbers. You can also ask the same question as how many non-perfect square numbers lie between two consecutive perfect square numbers? Can we come up with a general formula for that?
Lets write some perfect square numbers.
There are 2n natural numbers lying between two consecutive perfect square numbers
and
. We can see bunch of examples.
Lets choose two consecutive perfect square numbers. Lets say that we have chosen
and
. We can see that there are two numbers (2 and 3) which lie between
and
. Note that
If, we choose
and
. We can see that there are 4 numbers (5, 6, 7, 8) which lie between
and
. Note that
.
If, we choose
and
. We can see that there are 6 numbers (10, 11, 12, 13, 14, 15) which lie between
and
. Note that
How many natural numbers lie between
and
?
(A) 2257
(B) 2222
(C) 2356
(D) 2500
(E) None of these
We know that there are 2n natural numbers which lie between two consecutive perfect square numbers
and
. Therefore, between
and
, there are
natural numbers. Therefore, the answer is (B).
Practice Problems:
- How many natural numbers lie between
and
?
- How many non-perfect square numbers lie between
and
?
- Do you know how many natural numbers lie between any two given natural numbers? How many natural numbers are present between 1555 and 3520?
Answer: For any two given natural numbers n and m where n>m. There are (n-m-1) natural numbers between n and m. Therefore, between 1555 and 3520, there are 3520-1555-1=1964 natural numbers.
KRITHI KALYANI SIVAMURUGAN says
NICE
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