Page 42 - Start Up Mathematics_6

Patterns
Patterns in numbers are not only interesting, but are useful tools for verbal calculations. They also
help us to understand various properties in a better way.
Let’s observe the following pattern:
1 + 3 = 4 = 2 2

1 + 3 + 5 = 9 = 3 2
1 + 3 + 5 + 7 = 16 = 4 2
:

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8 2
We observe the sum of the first two odd numbers is equal to square of 2, the sum of the first three
odd numbers is equal to square of 3, the sum of the first four odd numbers is equal to square of
4, etc. In patterns like these, we may estimate the sum without actual calculations.
Example 13: Observe the pattern and fill in the blanks.

3 × 3 – 2 × 2 = 9 – 4 = 5 = 3 + 2
4 × 4 – 3 × 3 = 16 – 9 = 7 = 4 + 3
5 × 5 – 4 × 4 = 25 – 16 = 9 = 5 + 4

21 × 21 – 20 × 20 = 441 – 400 = 41 = 21 + 20
51 × 51 – 50 × 50 = 2,601 – 2,500 = 101 = 51 + 50
101 × 101 – 100 × 100 = ..............................................................................

525 × 525 – 524 × 524 = ..............................................................................
Solution: 101 × 101 – 100 × 100 = 101 + 100 = 201

525 × 525 – 524 × 524 = 525 + 524 = 1,049
Example 14: Study the pattern.
Puzzle
1 × 8 + 1 = 9 Each of the boxes in the given 3 × 3 grid are

12 × 8 + 2 = 98 filled by one of the digits 0, 1, 2, 3, 4, 5, 6, 7
and 8. Each letter represents a distinct digit.
123 × 8 + 3 = 987 If the products AEI, GEC, DEF A D G
Write the next two steps. Can you and BEH are equal, what digit B E H
explain how the pattern works? does E represent? C F I
Solution: The next two steps are as follows:

1,234 × 8 + 4 = 9,876
12,345 × 8 + 5 = 98,765 Do you know?

To know how it works, let’s observe the following: 1 GOOGOL = 10 100
12 × 8 + 2 = (11 + 1) × 8 + 2 = (88 + 8) + 2
= (88 + 1) + (8 + 1) = 89 + 9 = 98

1,234 × 8 + 4 = (1,111 + 111 + 11 + 1) × 8 + 4 = (8,888 + 888 + 88 + 8) + 4
= (8,888 + 1) + (888 + 1) + (88 + 1) + (8 + 1)

= 8,889 + 889 + 89 + 9 = 9,876

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