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Here, we observe that the sums of the odd numbers in the successive rows give the cubes of
numbers 1, 2, 3, 4 and so on.

2. Sum of the cubes of consecutive n numbers
The sum of the cubes of first n natural numbers is equal to the square of their sum, i.e.,
3
3
3
3
3
1 + 2 + 3 + 4 + … + n = (1 + 2 + 3 + 4 + … + n) 2
3
3
3
3
For example, 1 + 2 + 3 + 4 = (1 × 1 × 1) + (2 × 2 × 2) + (3 × 3 × 3) + (4 × 4 × 4)
= 1 + 8 + 27 + 64 = 100
2
and (1 + 2 + 3 + 4) = 10 = 100
2
3
Thus, 1 + 2 + 3 + 4 = (1 + 2 + 3 + 4) 2
3
3
3
In general, we can say that the sum of the cubes of n natural numbers is equal to the square
of sum of those n numbers i.e.,
(
 nn + )1 2
3
3
3
2
3
1 + 2 + 3 + ... + n = (1 + 2 + 3 + ... + n) =  
 2 
3. Difference of cubes of consecutive numbers
Observe the given pattern by taking the difference(s) of the cubes of consecutive numbers:
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2 – 1 = 8 – 1 = 7 = 1 + 6 = 1 + 3(2 × 1)
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3
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3 – 2 = 27 – 8 = 19 = 1 + 18 = 1 + 3(3 × 2)
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4 – 3 = 64 – 27 = 37 = 1 + 36 = 1 + 3(4 × 3)
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3
Thus, in general form, we can say that n – (n – 1) = 1 + 3n(n – 1).
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How to find the n consecutive odd numbers whose sum is equal to n ?
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• When n is odd: To find the consecutive n odd numbers, take the middle number as n , and then
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2
2
2
n – 2, n – 4, … on the left side and n + 2, n + 4, … on the right side.
• When n is even: To find consecutive n odd numbers, take the middle two numbers as n – 1 and
2
2
2
2
n + 1 of the given n. Then take n – 3, n – 5, … on the left side and n + 3, n + 5, … on the right
2
2
side.
Example 8: Express each of the following numbers as the sum of consecutive odd numbers.
(a) 6 3 (b) 9 3 (c) 11 3
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Solution: (a) Here, n = 6 (even); 6 = 31 + 33 + 35 + 37 + 39 + 41 = 216
3
(b) Here, n = 9 (odd); 9 = 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89 = 729
(c) Here, n = 11 (odd); 11 = 111 + 113 + 115 + 117 + 119 + 121 + 123 + 125 + 127 + 129 + 131
3
= 1331
3
3
Example 9: Using the relation n – (n – 1) = 1 + 3n(n – 1), find the value of each of the following:
(a) 8 – 7 3 (b) 12 – 11 3 (c) 100 – 99 3
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Solution: (a) 8 – 7 = 1 + 3 (8 × 7) = 169
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3
(b) 12 – 11 = 1 + 3 (12 × 11) = 397
3
3
(c) 100 – 99 = 1 + 3 (100 × 99) = 29701
3
3
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