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Some Other Interesting Patterns

Pattern 6. Squares of natural numbers having 1 as all its digits, follow the given pattern.

Sum of the digits
2
1 = 1 1 = 1 2
2
11 = 1 2 1 1 + 2 + 1 = 4 = 2 2
111 = 1 2 3 2 1 1 + 2 + 3 + 2 + 1 = 9 = 3 2
2
2
1111 = 1 2 3 4 3 2 1 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4 2
2
11111 = 1 2 3 4 5 4 3 2 1 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 = 5 2
111111 = 1 2 3 4 5 6 5 4 3 2 1 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 = 6 2
2
In the above pattern, the sum of the digits of every number on the right hand side is a perfect
square.
Pattern 7. The squares of the natural numbers like 11, 111 …, etc. have a special pattern as shown
below.
2
1 = 1
2
11 = 1 2 1
2
111 = 1 2 3 2 1
2
1111 = 1 2 3 4 3 2 1
2
Since, 11 = 121, now, 121 × (1 + 2 + 1) = 121 × 4 = 484 = 22 2
2
So, 11 × (sum of digits in 11 ) = (2 × 11) .
2
2
2
Further, 111 = 12321 and 12321 × (1 + 2 + 3 + 2 + 1) = 12321 × 9 = 110889 = 333 2
2
2
So, 111 × (sum of the digits in 111 ) = (3 × 111) 2
2
Again, 1111 = 1234321 and 1234321 × (1 + 2 + 3 + 4 + 3 + 2 + 1) = 1234321 × 16 = 19749136
= (4444) 2
2
2
So, 1111 × (sum of the digits in 1111 ) = (4 × 1111) 2
Pattern 8.
Observe the following pattern.
7 = 49
2
2
67 = 4489
2
667 = 444889
2
6667 = 44448889
2
66667 = 4444488889
666667 = 444444888889
2
2
6666667 = 44444448888889
2
2
2
We can find square of numbers like 66666667 , 666666667 and 6666666667 just using the above
pattern.
Example 13: Find the sum of the following numbers without adding.
(a) 1 + 3 + 5 + 7 + 9
(b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

(c) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23

131 Squares and Square Roots