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Without drawing a diagram, find:
(i) 10th square number:
A square number can easily be remembered by the following rule
Nth square number = n × n
10th square number = 10 × 10 = 100
(ii) 6th triangular number:
A triangular number can easily be remembered by the following rule
Nth triangular number = n × ( n + 1 )2
Therefore, 6th triangular number = 6 × (6 + 1)2 = 21
(i) Can a rectangle number also be a square number?
(ii) Can a triangular number also be a square number?
(i) Yes, a rectangular number can also be a square number; for example, 16 is a square number also a rectangular number.
(ii) Yes, there e×ists only one triangular number that is both a triangular number and a square number, and that number is 1.
Write the first four products of two numbers with difference 4 starting from in the following order:
1 , 2 , 3 , 4 , 5 , 6 , ………..
Identify the pattern in the products and write the next three products.
1 × 5 = 5 (5 – 1 = 4)
2 × 6 = 12 (6 – 2 =4)
3 × 7 = 21 (7 – 3 = 4)
4 × 8 = 32 (8 – 4 = 4)
Observe the pattern in the following and fill in the blanks:
9 × 9 + 7 =88
98 × 9 + 6 = 888
987 × 9 + 5 = 8888
9876 × 9 + 4 = 88888
98765 × 9 + 3 = 888888
987654 × 9 + 2 = 8888888
9876543 × 9 + 1 = 88888888
Observe the following pattern and extend it to three more steps:
6 × 2 – 5 = 7
7 × 3 – 12 = 9
8 × 4 – 21 = 11
9 × 5 – 32 = 13
10 × 6 – 45 = 15
11 × 7 – 60 = 17
12 × 8 – 77 = 19
Study the following pattern:
1 + 3 = 2 × 2
1 + 3 + 5 = 3 × 3
1 + 3 + 5 + 7 = 4 × 4
1 + 3 + 5 + 7 + 9 = 5 × 5
By observing the above pattern, find:
(i) 1 + 3 + 5 + 7 + 9 + 11
= 6 × 6
= 36
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15
= 8 × 8
= 64
(iii) 21 + 23 + 25 + … + 51
= (21 + 23 + 25 +…+ 51) can also be written as
(1 + 3 + 5 + 7 +…+ 49 + 51) – (1 + 3 + 5 +…+ 17 + 19)
(1 + 3 + 5 + 7+…+ 49 + 51) = 26 × 26 = 676
and, (1 + 3 + 5 +…+ 17 + 19 ) = 10 × 10 = 100
Now,
(21 + 23 + 25 +…+ 51 ) = 676 – 100 = 576
By observing the above pattern, write next two steps.
The next two steps are as follows:
1 × 1 + 2 × 2 + 3 × 3 + 4 × 4 + 5 × 5
= 5 × 6 × 116
= 55
1 × 1+ 2 × 2 + 3 × 3 + 4 × 4 + 5 × 5 + 6 × 6
= 6 × 7 × 136
= 91
(i) 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
= 10 × 112
(ii) 50 + 51 + 52 + …+ 100
This can also be written as
(1 + 2 + 3 + …+ 99 + 100) – (1 + 2 + 3 + 4 + …+ 47 + 49)
(1 + 2 + 3 + …+ 99 + 100 ) = 100 × 1012
and, (1 + 2 + 3 + 4 +…+ 47 + 49 ) = 49 × 502
So, (50 + 51 + 52 + …+ 100 ) = 100 × 1012 – 49 × 502
= 5050 – 1225
= 3825
(iii) 2 + 4 + 6 + 8 + 10 +…+ 100
This can also be written as 2 × (1 + 2 + 3 + 4 + …+ 49 + 50)
(1 + 2 + 3 + 4 + …+ 49 + 50 ) = 50 × 512
= 1275
Therefore, (2 + 4 + 6 + 8 + 10 + …+ 100) = 2 × 1275 = 2550
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Chapter 4: Operations on Whole Numbers –...