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SQUARES AND SQUARE ROOTS 93

If we combine two consecutive triangular numbers, we get a square number, like

1 + 3 = 4 3 + 6 = 9 6 + 10 = 16
= 2 2 = 3 2 = 4 2

2. Numbers between square numbers
Let us now see if we can find some interesting pattern between two consecutive
square numbers.

Two non square numbers
2
1 (= 1 )
6 non square numbers between between the two square
2
2
2
the two square numbers 9(=3 ) numbers 1 (=1 ) and 4(=2 ).
and 16(= 4 ). 2, 3, 4 (= 2 )
2
2
2
8 non square 5, 6, 7, 8, 9 (= 3 )
numbers between
the two square 4 non square numbers
2
2
numbers 16(= 4 ) 10, 11, 12, 13, 14, 15, 16 (= 4 ) between the two square
2
and 25(=5 ).
2
2
numbers 4(=2 ) and 9(3 ).
2
17, 18, 19, 20, 21, 22, 23, 24, 25 (= 5 )
2
Between 1 (=1) and 2 (= 4) there are two (i.e., 2 × 1) non square numbers 2, 3.
2
2
2
Between 2 (= 4) and 3 (= 9) there are four (i.e., 2 × 2) non square numbers 5, 6, 7, 8.
2
2
Now, 3 = 9, 4 = 16
Therefore, 4 – 3 = 16 – 9 = 7
2
2
2
2
Between 9(=3 ) and 16(= 4 ) the numbers are 10, 11, 12, 13, 14, 15 that is, six
non-square numbers which is 1 less than the difference of two squares.
2
2
We have 4 = 16 and 5 = 25
2
2
Therefore, 5 – 4 = 9
Between 16(= 4 ) and 25(= 5 ) the numbers are 17, 18, ... , 24 that is, eight non square
2
2
numbers which is 1 less than the difference of two squares.
2
2
2
Consider 7 and 6 . Can you say how many numbers are there between 6 and 7 ?
2
If we think of any natural number n and (n + 1), then,
2
2
2
2
(n + 1) – n = (n + 2n + 1) – n = 2n + 1.
2
2
We find that between n and (n + 1) there are 2n numbers which is 1 less than the
difference of two squares.
Thus, in general we can say that there are 2n non perfect square numbers between
the squares of the numbers n and (n + 1). Check for n = 5, n = 6 etc., and verify.