Chapter 4 – Patterns

📝 Practice Set 4.1 📝 Practice Set 4.2 📝 Exercise 4

Exercise 4

(1) Complete the table below.


2	5	10	17

Solution

Observe the pattern in the number of dots:

1² + 1 = 2; 2² + 1 = 5; 3² + 1 = 10; 4² + 1 = 17; 5² + 1 = 26; 6² + 1 = 37.


2	5	10	17	26	37

(2) The figures showing hexagonal numbers are drawn below. Draw them in your notebook. What is the next number in this figure? Show it with the help of the figure.

Solution

Observe the pattern in the figures.

1 + 6 = 7
1 + 6 + 12 = 19
1 + 6 + 12 + 18 = 37
1 + 6 + 12 + 18 + 24 = 61

The next figure has 61 dots:

Ans. The next number in the figure is 61.

(3) Observe the number pattern and write the next three numbers.
2, 9, 28, 65, ..., ..., ...

Solution

2 , 9 , 28 , 65 , 126 , 217 , 344

Hint

1³ + 1 = 2; 2³ + 1 = 9; 3³ + 1 = 28; 4³ + 1 = 65; 5³ + 1 = 126; 6³ + 1 = 217; 7³ + 1 = 344.

Ans. 126, 217, 344.

(4) Write the first five triangular numbers. What is the tenth triangular number?

Solution

First triangular number = 1

Second triangular number = 1 + 2 = 3

Third triangular number = 1 + 2 + 3 = 6

Fourth triangular number = 1 + 2 + 3 + 4 = 10

Fifth triangular number = 1 + 2 + 3 + 4 + 5 = 15

First five triangular numbers:

1, 3, 6, 10, 15

Tenth triangular number = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55.

Ans. First five triangular numbers are 1, 3, 6, 10, 15. The tenth triangular number is 55.

(5) What number series is formed if the counting numbers are added in ascending order?

Solution

If counting numbers are added in ascending order, triangular numbers series is formed.

Example:
1 = 1;
1 + 2 = 3;
1 + 2 + 3 = 6;
1 + 2 + 3 + 4 = 10.

Ans. Triangular numbers series is formed.

(6) Write the sequence of numbers formed by adding the odd numbers starting from 1.

(Ex. Square numbers: 1, 1+3, 1+3+5,..........)

Solution

1	1² = 1
1 + 3 = 4	2² = 4
1 + 3 + 5 = 9	3² = 9
1 + 3 + 5 + 7 = 16	4² = 16

The sequence of numbers formed by adding consecutive odd numbers starting from 1 are square numbers.

Ans. The sequence is 1, 4, 9, 16, ... and these are square numbers.

(7) What is the sum of 1+2+3 + ... + 50 + 49 +...+3+2+1?

Solution

Ans. 2500

Hint

1 + 2 + 3 + ··· + 50 + 49 + ··· + 2 + 1

Using the formula:

1 + 2 + 3 + ··· + n = n(n + 1) 2

For 1 to 50, n = 50:

50(51) 2 = 1275

The numbers from 49 to 1 are in reverse order, so it is the same as 1 to 49. Here, n = 49:

49(50) 2 = 1225

Therefore,

1275 + 1225 = 2500

Chapter 4 – Patterns – Exercise 4

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