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94 MATHEMATICS

TRY THESE
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1. How many natural numbers lie between 9 and 10 ? Between 11 and 12 ?
2. How many non square numbers lie between the following pairs of numbers
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(i) 100 and 101 2 (ii) 90 and 91 2 (iii) 1000 and 1001 2
3. Adding odd numbers
Consider the following
1 [one odd number] = 1 = 1 2
1 + 3 [sum of first two odd numbers] = 4 = 2 2
1 + 3 + 5 [sum of first three odd numbers] = 9 = 3 2
1 + 3 + 5 + 7 [... ] = 16 = 4 2
1 + 3 + 5 + 7 + 9 [... ] = 25 = 5 2
1 + 3 + 5 + 7 + 9 + 11 [... ] = 36 = 6 2
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So we can say that the sum of first n odd natural numbers is n .
Looking at it in a different way, we can say: ‘If the number is a square number, it has
to be the sum of successive odd numbers starting from 1.
Consider those numbers which are not perfect squares, say 2, 3, 5, 6, ... . Can you
express these numbers as a sum of successive odd natural numbers beginning from 1?
You will find that these numbers cannot be expressed in this form.

Consider the number 25. Successively subtract 1, 3, 5, 7, 9, ... from it
(i) 25 – 1 = 24 (ii) 24 – 3 = 21 (iii) 21 – 5 = 16 (iv) 16 – 7 = 9
(v) 9 – 9 = 0
This means, 25 = 1 + 3 + 5 + 7 + 9. Also, 25 is a perfect square.
Now consider another number 38, and again do as above.

(i) 38 – 1 = 37 (ii) 37 – 3 = 34 (iii) 34 – 5 = 29 (iv) 29 – 7 = 22
(v) 22 – 9 = 13 (vi) 13 – 11 = 2 (vii) 2 – 13 = – 11
This shows that we are not able to express 38 as the
TRY THESE sum of consecutive odd numbers starting with 1. Also, 38 is
not a perfect square.
Find whether each of the following
So we can also say that if a natural number cannot be
numbers is a perfect square or not?
expressed as a sum of successive odd natural numbers
(i) 121 (ii) 55 (iii) 81
starting with 1, then it is not a perfect square.
(iv) 49 (v) 69
We can use this result to find whether a number is a perfect
square or not.
4. A sum of consecutive natural numbers
Consider the following
First Number 3 = 9 = 4 + 5 Second Number
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2 2
3 − 1 2 3 + 1
= 5 = 25 = 12 + 13 =
2 2
7 = 49 = 24 + 25
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