Page 43 - ICSE Math 8

Addition of two consecutive triangular numbers forms a square number.

1 + 3 = 4 = 2 2 3 + 6 = 9 = 3 2 6 + 10 = 16 = 4 2
2
2
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 two 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
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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
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IV. If two consecutive odd natural numbers are multiplied and 1 is added to the product, it is equal to the
square of the even number between them, i.e., {(2n – 1) × (2n + 1)} + 1 = (2n) 2

2
2
For example, 1 × 3 + 1 = 4 = 2 3 × 5 + 1 = 16 = 4 5 × 7 + 1 = 36 = 6 2
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
2
2
2
2
2
numbers, i.e., (2n + 1) = 4n + 4n + 1 {Q (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. The natural numbers containing all digits as 6 with units digit as 7 follow another interesting pattern.
2
7 = 49
2
67 = 4489
2
667 = 444889
2
6667 = 44448889 and so on.
Example 4: Which of the following natural numbers are not perfect squares? Give reasons.
(a) 2,037 (b) 1,024 (c) 5,298 (d) 33,222 (e) 13,456
Solution: (a), (c) and (d) are not perfect squares because numbers ending with digits 2, 3, 7, 8 are never
perfect squares.
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