Page 56 - Start Up Mathematics_8 (Non CCE)

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II. In between two consecutive square numbers n and (n + 1) , there exists ‘2n’ non-perfect square numbers.
Consecutive Non-perfect square numbers between Number of non-perfect
square numbers 2 consecutive square numbers square numbers
2
1 , 2 2 2, 3 2 = 2 × 1
2
2 , 3 2 5, 6, 7, 8 4 = 2 × 2
2
3 , 4 2 10, 11, 12, 13, 14, 15 6 = 2 × 3
  
2
2
2
2
2
n , (n + 1) 2 n + 1, n + 2, n + 3, ..., n + 2n 2n = 2 × n
III. Squares of natural numbers having all digits as 9 follow the given pattern.
Number Squares Pattern
2
9 9 = 81 8 + 1 = 9 = 9 × 1
2
99 99 = 9801 9 + 8 + 0 + 1 = 18 = 9 × 2
2
999 999 = 998001 9 + 9 + 8 + 0 + 0 + 1 = 27 = 9 × 3
2
9999 9999 = 99980001 9 + 9 + 9 + 8 + 0 + 0 + 0 + 1 = 36 = 9 × 4
and so on.

IV. If two consecutive odd natural numbers are multiplied Remember
and 1 is added to the product, it is equal to the square
of the even number between them, Square of any natural number consisting of
i.e., {(2n – 1) × (2n + 1)} + 1 = (2n) 2 only ‘n’ number of nines can be written as
2
For example, 1 × 3 + 1 = 4 = 2 (9999 ... n times) 2
2
3 × 5 + 1 = 16 = 4 5 × 7 + 1 = 36 = 6 2 = (n – 1) nines 8 (n – 1) zeros 1
V. If two consecutive even natural numbers are multiplied
and 1 is added to the product, it is equal to the square
of the odd number between them, i.e., {(2n) × (2n + 2)} + 1 = (2n – 1) 2
2
2
For example, 2 × 4 + 1 = 9 = 3 4 × 6 + 1 = 25 = 5 6 × 8 + 1 = 49 = 7 2
VI. The square of any odd natural number other than 1 can be represented as a sum of 2 consecutive natural
numbers.
2
2
2
2
2
i.e., (2n + 1) = 4n + 4n + 1 { (a + b) = a + b + 2ab}
2
2
2
2
= 2n + 2n + 2n + 2n + 1 = (2n + 2n) + (2n + 2n + 1)
2
2
2
For example, 3 = 9 = 4 + 5 5 = 25 = 12 + 13 7 = 49 = 24 + 25
VII. Squares of natural numbers having all digits as 1 follow an interesting pattern.
Number Squares Pattern
2
11 (11) = 121 1 + 2 + 1 = 4 = 2 2
2
111 (111) = 12321 1 + 2 + 3 + 2 + 1 = 9 = 3 2
2
1111 (1111) = 1234321 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4 2
  
2
111111111 (111111111) = 12345678987654321 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 8 + 7 + 6
2
+ 5 + 4 + 3 + 2 + 1 = 81 = 9
This pattern is restricted upto natural numbers having nine 1’s.
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