Page 7 - NCERT Books

SQUARES AND SQUARE ROOTS 95

2
9 = 81 = 40 + 41
Vow! we can express the
2
11 = 121 = 60 + 61
square of any odd number as
2
15 = 225 = 112 + 113 the sum of two consecutive
positive integers.
TRY THESE
1. Express the following as the sum of two consecutive integers.
(i) 21 2 (ii) 13 2 (iii) 11 2 (iv) 19 2
2. Do you think the reverse is also true, i.e., is the sum of any two consecutive positive
integers is perfect square of a number? Give example to support your answer.

5. Product of two consecutive even or odd natural numbers
2
11 × 13 = 143 = 12 – 1
Also 11 × 13 = (12 – 1) × (12 + 1)
Therefore, 11 × 13 = (12 – 1) × (12 + 1) = 12 – 1
2
2
Similarly, 13 × 15 = (14 – 1) × (14 + 1) = 14 – 1
29 × 31 = (30 – 1) × (30 + 1) = 30 – 1
2
2
44 × 46 = (45 – 1) × (45 + 1) = 45 – 1
2
So in general we can say that (a + 1) × (a – 1) = a – 1.
6. Some more patterns in square numbers

Observe the squares of numbers; 1, 11, 111 ... etc. They give a beautiful pattern:
2
1 = 1
2
11 = 1 2 1
2
111 = 1 2 3 2 1
2
1111 = 1 2 3 4 3 2 1
11111 = 1 2 3 4 5 4 3 2 1
2
2
11111111 = 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1
Another interesting pattern. TRY THESE

2
7 = 49
Write the square, making use of the above
67 = 4489
2
pattern.
2
667 = 444889 (i) 111111 2 (ii) 1111111 2
2
6667 = 44448889
66667 = 4444488889 TRY THESE
2
2
666667 = 444444888889
Can you find the square of the following
The fun is in being able to find out why this happens. May
numbers using the above pattern?
be it would be interesting for you to explore and think about
(i) 6666667 2 (ii) 66666667 2
such questions even if the answers come some years later.