RBSE Class 6 Maths Chapter 1 Patterns in Mathematics Solutions

Practicing RBSE Class 6 Maths Solutions and Class 6 Maths Chapter 1 Patterns in Mathematics Solutions Question Answer helps develop logical thinking and accuracy.

Patterns in Mathematics Class 6 Solutions

Ganita Prakash Class 6 Chapter 1 Solutions Patterns in Mathematics

Figure it Out (Page 2)

Question 1.
Can you think of other examples where mathematics helps us in our everyday lives?
Solution:
Mathematics helps us in our everyday life very much, in which some examples are given below—

1. In Shopping : When we buy goods at a shop, we have to calculate how much money we have to pay and how much we have to take.
2. In Time Management : Mathematics is used to read clocks and keep track of time.
3. Measurements and Sizes : Mathematics is used to measure the length, width and height of any object.
4. In Sports : Keeping score while playing requires math for measuring distances and keeping records.
5. In Accounts : We use maths to understand our pocket money and save money.

Question 2.
How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.)
Solution:
Mathematics has helped to advance and inspire humanity in many important ways, some of which are as follows—

1. Development of problem solving ability
2. Development of science and technology
3. Invention and Innovations
4. Financial Management
5. Contribution in the building of modem society
6. Education and Intellectual development.

Figure it Out (Page 3)

Question 1.
Can you recognise the pattern in each of the sequences in Table 1?
(a) 1, 1, 1, 1, 1, 1, 1, ………………… (All 1’s)
(b) 1, 2, 3, 4, 5, 6, 7, ………………… (Counting numbers)
(c) 1, 3, 5, 7, 9, 11, 13, ………………… (Odd numbers)
(d) 2, 4, 6, 8, 10, 12, 14, ………………… (Even numbers)
(e) 1, 3, 6, 10, 15, 21, 28, ………………… (Triangular numbers)
(f) 1, 4, 9, 16, 25, 36, 49, ………………… (Squares)
(g) 1, 8, 27, 64, 125, 216, ………………… (Cubes)
(h) 1, 2, 3, 5, 8, 13, 21, ………………… (Virah\(\bar{a}\)nka numbers)
(i) 1, 2, 4, 8, 16, 32, 64, ………………… (Powers of 2)
(j) 1, 3, 9, 27, 81, 243, 729, ………………… (Powers of 3)
Solution:
(a) Repetition of number ‘1′.

(b) Counting numbers starting from 1 : 1, 1 + 1, 1 + 1 + 1, …………………….

(c) Odd numbers starting from 1 : 1, 1 + 2 = 3, 3 + 2 = 5, 5 + 2 = 7, …………………….

(d) Even numbers starting from 2 : 2, 2 + 2 = 4, 4 + 2 = 6, 6 + 2 = 8, 8 + 2 = 10, …………………….

(e) Triangular numbers:
1 = 1
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
……………………
……………………

(f) Square numbers : Squares of counting numbers :
1 = 1 × 1
4 = 2 × 2
9 = 3 × 3
16 = 4 × 4
25 = 5 × 5
……………………
……………………

(g) Cubes :
1³ = 1 × 1 × 1 = 1
2³ = 2 × 2 × 2 = 8
3³ = 3 × 3 × 3 = 27
4³ = 4 × 4 × 4 = 64
……………………………………………….
………………………………………………

(h) Virah\(\bar{a}\)nka numbers : (Sum of last two numbers)
1, 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, ………………………..

(i) Powers of 2 :
2⁰ = 1, 2¹ = 2
2² = 2 × 2 = 4
2³ = 2 × 2 × 2 = 8
2⁴ = 2 × 2 × 2 × 2 = 16
……………………………………………………………….

(j) Powers of 3 :
3⁰ = 1, 3¹ = 3
3² = 3 × 3 = 9
3³ = 3 × 3 × 3 = 27
3⁴ = 3 × 3 × 3 × 3 = 81
3⁵ = 3 × 3 × 3 × 3 × 3 = 243
3⁶ = 3 × 3 × 3 × 3 × 3 × 3 = 729
………………………………………………………………………………

Question 2.
Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.
Solution:

Figure it Out (Page 5)

Question 1.
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
Table 2—Pictorial representation of some number sequences

Solution:

Question 2.
Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes?
Solution:
1, 3, 6, 10, 15, ………………….. are called triangular numbers because these numbers of dots can be arranged to make a triangle.

1, 4, 9, 16, 25, ………………….. are called square numbers or squares because these numbers of dots can be arranged to make a square.

1, 8, 27, 64, 125, ………………….. are called cube numbers or cubes because 1,8, 27, 64, 125, ………………….. units can be arranged to make cubes.

Question 3.
You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!
Solution:

Other similar numbers are : 1, 36, 1225, 41616, 1413721, 48024900,1631432881, 55420693056, 1882672131025, 63955431761796

Question 4.
What would you call the following sequence of numbers?

That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?
Solution:
First hexagonal number = 1
Second hexagonal number = 1 + 6 = 7
Third hexagonal number = 7 + 6 × 2 = 7 + 12 = 19
Fourth hexagonal number = 19 + 6 × 3 = 19 + 18 = 37
Fifth hexagonal number = 37 + 6 × 4 = 37 + 24 = 61
Hence the next number in the sequence is 61.

Question 5.
Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?
Solution:
Here is one possible way of thinking about powers of 2.

Here is one possible way thinking about powers of 3.

3³ = 27 (Twenty seven concentric circles)

Figure it Out (Page 8)

Question 1.
Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 +1, …, gives square numbers?
Solution:

This pattern gives square numbers because we can see the arrangements of terms in each sum.
1 = 1 = (1)²
1 + 2 + 1 = 4 = (2)²
1 + 2 + 3 + 2 + 1 = 9 = (3)²
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = (4)²
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 = (5)²
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = (6)²
Clearly, we can observe that each term is symmetric.

Question 2.
By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of 1 + 2 + 3 + … + 99 + 100 + 99 + … + 3 + 2 + 1?
Solution:
1 + 2 + 3 + …. + 99 + 100 + 99 + …. + 3 + 2 + 1 = 100 × 100 = 10000

Question 3.
Which sequence do you get when you start to add the All 1’s sequence up? What sequence do you get when you add the All 1’s sequence up and down?
Solution:
When we start to add the all l’s sequence up, we get the following sequence—
1, 1, 1, 1, 1, 1, ……………………..
We get the same sequence when we add the all 1’s sequence up and down. For example, when all five 1 ’s sequence up: 1 + 1 + 1 + 1 + 1 = 5 when all five 1’s sequence down : 1 + 1 + 1 + 1 + 1 = 5

Question 4.
Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
Solution:
Adding the counting numbers up
1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
…………………………….
…………………………….
Required sequence
1, 3, 6, 10, …………………………….
Pictorial explanation:

Question 5.
What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … Which sequence do you get? Why? Can you explain it with a picture?
Solution:
Consecutive triangular numbers are :
1, 3, 6, 10, 15, 21, 28, …………………………
When we add up pairs of these numbers—
1 + 3 = 4 = 2 × 2
3 + 6 = 9 = 3 × 3
6 + 10 = 16 = 4 × 4
Hence, we get square numbers when we add up pairs of consecutive triangular numbers.
Required sequence—4, 9, 16, 36, …………………………
Pictorial explanation:

Question 6.
What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, … ? Now add 1 to each of these numbers—what numbers do you get? Why does this happen?
Solution:
Adding up powers of 2 starting with 1

Hence, the required sequence is 1, 3, 7, 15, 31, …………………..
Adding 1 to each of these numbers—

Hence, we get the sequence 2, 4, 8, 16, …………………. increasing multiples of 2. This pattern occurs because it is a property of geometric series where she sum of the first terms of the series after doubling each time is just one less than the next power, i.e. 1 + 2 + 4 + 8 + 2ⁿ = ………………….. 2^{n + 1} – 1.

Question 7.
What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?
Solution:
Triangular numbers are—
1, 3, 6, 10, 15, 21, ………………………….
We multiply die triangular numbers by 6 and add 1.
1 × 6 + 1 = 7
3 × 6 + 1 = 19 (Increase of 12 i.e. 6 × 2)
6 × 6 + 1 = 37 (Increase of 18 i.e. 6 × 3)
10 × 6 + 1 = 61 (Increase of 24 i.e. 6 × 4)
15 × 6 + 1 = 91 (Increase of 30 i.e. 6 × 5)

Required sequence : 7, 19, 37, 61, 91, ……………………

Question 8.
What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, … ? Which sequence do you get? Can you explain it using a picture of a cube?

Solution:
Hexagonal numbers are
1, 7, 19, 37, 61, ………………………
According to question,
1 = 1 = 1 × 1 × 1 = 1³
1 + 7 = 8 = 2 × 2 × 2 = 2³
1 + 7 + 19 = 27 = 3 × 3 × 3 = 3³
1 + 7 + 19 + 37 = 64 = 4 × 4 × 4 = 4³
1 + 7 + 19 + 37 + 61 = 125 = 5 × 5 × 5 = 5³
Figure:

Question 9.
Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
Solution:
(1) First sequence 1, 1, 2, 3, 5, 8, …………………

(2) Second sequence : 1, 5, 13, 25, 41, ……………………..

Figure it Out (Page 11)

Question 1.
Can you recognise the pattern in each of the sequences in Table 3?


Solution:
(a) Triangle = 3 sides
Quadrilateral = 4 sides
Pentagon = 5 sides
Hexagon = 6 sides
Heptagon = 7 sides
Octagon = 8 sides
Nonagon = 9 sides
Decagon = 10 sides.
Hence, the required sequence is : 3, 4, 5, 6, 7, 8, 9, 10, …………………….
It is a continuous number sequence starting from 3.

(b) K₂ = 1 = 1
K₃ = 3 = 1 + 2
K₄ = 6 = 1 + 2 + 3
K₅ = 10 = 1 + 2 + 3 + 4
K₆ = 15 = 1 + 2 + 3 + 4 + 5
Hence, the sequence is : 1, 3, 6, 10, 15, ………………………
This is a triangular number sequence.

(c) Here, the number of squares are : 1, 4, 9, 16, 25, ………………………
1 = 1 × 1
4 = 2 × 2
9 = 3 × 3
16 = 4 × 4
25 = 5 × 5
Hence, it is a presentation of square number sequence.

(d) Number of small triangles in each :
1 = 1 × 1
4 = 2 × 2
9 = 3 × 3
16 = 4 × 4
25 = 5 × 5
Hence, the sequence is : 1, 4, 9, 16, 25, …………………..
Hence, it is also square number sequence shown by triangles.

(e) The sequence is : 3, 12, 48, 192, 768, …………………………
Here the number of sides becomes four times.

Question 2.
Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
Solution:
(a) Regular Polygon : Polygon of 11 sides. (Hendecagon)

(d) Stacked Triangles

Total number of triangle = 1 + 3 + 5 + 7 + 9 + 11 = 36

(e) Koch Snowflake : 768 × 4 = 3072, It is not possible to draw the snowflake with so many sides. Each sequence has been explained in question (1) as to which shapes are formed in that sequence.

Figure it Out (Page 11)

Question 1.
Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Solution:

Both sequences are same because in a regular polygon number of sides is equal to number of vertices.

Question 2.
Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
Solution:

Graphs	K2	K3	K4	K5	K6
Number of lines	1	3	6	10	15

Hence, we get number sequence: 1, 3, 6, 10, 15, ……………………
It is a triangular number sequence.

Question 3.
How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain, why?
Solution:

No. of small squares on each side	1	2	3	4	5	6	7
Total no. of small squares	1	4	9	16	25	36	49

We get the sequence: 1, 4, 9, 16, 25, 36, ………………………
Hence, we get sequence of square numbers because each term represents the area of square corresponding to each side.

Question 4.
How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in the sequence, how many triangles are there in each row?)
Solution:

No. of rows of small triangles	1	2	3	4	5	6	7
Total no. of small triangles	1	4	9	16	25	36	49

Sequence : 1, 4, 9, 16, 25, 36, 49, ………………….
These are square numbers. Hence by adding a stacked triangle at the bottom, next number in the square sequence will come.

Question 5.
To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speed bump’ . As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48, …, i.e., 3 times Powers of 4; this sequence is not shown in Table 1.)
Solution:

Koch Snowflake	1	2	3	4	5
No. of lines	3	12	48	192	768

Sequence: 3, 12, 48, 192, 768, ………………………..
Here each term can be obtained by multiplying the previous term by 4.

Patterns in Mathematics Class 6 Question Answer

Patterns in Mathematics Class 6 Extra Questions

Multiple Choice Questions—

Question 1.
What is the next number in the sequence
6, 18, 30, 42, 54, ………………….?
(a) 60
(b) 72
(c) 66
(d) 84
Answer:
(c) 66

Question 2.
The next number in the sequence 7, 10, 13, 16, 19 is—
(a) 21
(b) 20
(c) 22
(d) 23
Answer:
(c) 22

Question 3.
The number of sides in decagon is—
(a) 5
(b) 6
(c) 9
(d) 10
Answer:
(d) 10

Question 4.
The number of angles in nonagon is—
(a) 5
(b) 6
(c) 9
(d) 10
Answer:
(c) 9

Question 5.
Virah\(\overline{\mathrm{a}}\)nka numbers are—
(a) 1, 8, 27, 64, 125
(b) 1, 2, 3, 5, 8, 13, 21
(c) 1, 2, 4, 8, 16, 32
(d) 1, 3, 9, 27, 81, 243
Answer:
(b) 1, 2, 3, 5, 8, 13, 21

Question 6.
The next number in the sequence 2, 3, 5, 7, 11, 13 is—
(a) 17
(b) 15
(c) 16
(d) 14
Answer:
(a) 17

Question 7.
Which of the following numbers will complete the pattern?
1, 5, 9, 13, ?
(a) 14
(b) 15
(c) 17
(d) 18
Answer:
(c) 17

Question 8.
The next number in the sequence 2, 6, 12, 20, ……………………… is—
(a) 28
(b) 24
(c) 30
(d) 32
Answer:
(c) 30

Question 9.
The next figure of the given pattern is—

(a) □
(b) ■■
(c) ■
(d) □□
Answer:
(c) ■

Question 10.
The next number in the sequence 100, 95, 90, 85, …………………. is—
(a) 70
(b) 80
(c) 75
(d) 78
Answer:
(b) 80

Fill in the blanks—

1. The branch of Mathematics that studies patterns in whole numbers is called ………………………. .
Answer:
Number theory

2. Regular polygons, complete graphs, stacked triangles and squares are the examples of …………………… sequences.
Answer:
shape

3. Shape sequences exhibit many interesting relationships with …………………… sequences.
Answer:
number

4. 1, 3, 6, 10, 15, 21, 28, …………………… are …………………… numbers.
Answer:
triangular

5. The number …………………… is a square number as well as triangular number.
Answer:
36

6. The sum of the numbers of sides and angles in a pentagon is …………………… .
Answer:
10

Write True/False for the following statements—

1. Adding counting numbers up and down i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …………………… gives square
numbers. (True/False)
2. 1, 2, 3, 5, 8, 13, 21, ………………………. are triangular numbers. (True/False)
3. Each number in Fibonacci sequence is the sum of previous two numbers. (True/False)
4. 1, 5, 12, 22, 35, 51, …………………. are square (True/False)
Answer:
1. True
2. False
3. True
4. False

Make the right match—

Question 1.

(1) 1, 5, 12, 22, 35, 51, ……….	(a) Square Numbers
(2) 1, 4, 9, 16, 25, 36, 49, ……….	(b) Palindromic Numbers
(3) 1, 6, 15, 28, 45, 66, ……….	(c) Pentagonal Numbers
(4) 1, 11,121,1331, 14641, ……….	(d) Hexagonal Numbers

Answer:
(1) – (c), (2) – (a), (3) – (d), (4) – (b).

(1) 1, 5, 12, 22, 35, 51, ……….	(c) Pentagonal Numbers
(2) 1, 4, 9, 16, 25, 36, 49, ……….	(a) Square Numbers
(3) 1, 6, 15, 28, 45, 66, ……….	(d) Hexagonal Numbers
(4) 1, 11,121,1331, 14641, ……….	(b) Palindromic Numbers

Very Short Answer Type Questions—

Question 1.
What is the combination pattern of numbers?
Solution:
Here addition and subtraction are combined in the same pattern.
For example:
2, 5, 3, 6, 4, 7, 5, 8, …………………….
(2, 2 + 3, 5 – 2, 3 + 3, 6 – 2, 4 + 3, 7 – 2, 5 + 3, …………….)

Question 2.
What is the pattern of palindromic numbers?
Solution:
Palindromic numbers are those that remain the same even when read backwards.
1, 11, 121, 1331, 14641, …………….

Question 3.
Write the sequence of centered square numbers.
Solution:
1, 5, 13, 25, …………….

Question 4.
What is the pattern of dedicatory numbers?
Solution:
In the pattern of dedicative numbers, each number increases at a certain interval such that 7 is added to every number
1, 8, 15, 22, 29, 36, …………….

Question 5.
What is the next number in the given sequence?
5, 25, 125, 625, …………….
Solution:
3125 ( = 625 × 5)

Question 6.
What is the next number in the sequence 3, 6, 12, 24, …………….?
Solution:
48 (Every number is multipled by 2)

Question 7.
What is point illustration?
Solution:
When a sequence contains a particular group of number that are represented by dots, it is called a dot plot.

Question 8.
State the centered triangular sequence.
Solution:
1, 4, 10, 19, …………….

Question 9.
Write the sequence of centered octagonal numbers.
Solution:
1, 9, 25, 49, 81, 121, …………….

Question 10.
Write the next three numbers in the sequence 0.5, 1, 1.5, 2, 2.5, …………….
Solution:
3, 3.5, 4

Short Answer Type Questions—

Question 1.
What is the visualization of number sequences?
Solution:
Presenting a number sequence in a visual form so that it is easier to understand is called visualization of number sequences. For example—

1. Dot or Point illustration
2. Line illustration
3. Arrange in a grid
4. Geometrical shapes
5. Line chart
6. Spiral pattern
7. Colour coding

Question 2.
Give the relation in the number sequence 3, 6, 12, 24, 48, …………….
Solution:
3, 6, 12, 24, 48, …………….
This sequence is of numbers where each number is multiplied by 2 to get the next number. It can be understand as follows :
First term = 3
Second term = 3 × 2 = 6
Third term = 6 × 2 = 12
Fourth term = 12 × 2 = 24
Fifth term = 24 × 2 = 48

Question 3.
Give the relation in the number sequence 64, 32, 16, 8, 4, ……………..
Solution:
The sequence is of numbers where each number is divided by 2. It can be understand as follows :
First term: 64
Second term : 64 ÷ 2 = 32
Third term : 32 ÷ 2 = 16
Fourth term : 16 ÷ 2 = 8
Fifth term : 8 ÷ 2 = 4

Question 4.
Explain Fibonacci sequence.
Solution:
The number sequence 1, 1, 2, 3, 5, 8, 13, ………………. is a Fibonacci sequence, where each number (except first term) is the sum of previous two numbers.
First term: 1
Second term : 1
Third term : 1 + 1 = 2
Fourth term : 1 + 2 = 3
Fifth term : 2 + 3 = 5
Sixth term : 3 + 5 = 8
Seventh term : 5 + 8 = 13
and so on.

Essay Type Questions—

Question 1.
When the circles are arranged in a hexagonal shape, then what is the total number of small circles in each shape? What kind of sequence is formed? Can you explain why this is so?
Solution:

Number of layers of small circles	1	2	3	4	5	6	7
Number of small circles	1	7	19	37	61	91	127

Sequence : 1, 7, 19, 37, 61, 91, 127 …………………
This is a central hexagonal number sequence. The number of small circles increases in each layer such that it add up to a hexagonal shape.
1 = 1
7 = 1 + 6 × 1 = 7
19 = 7 + 6 × 2 = 19
37 = 19 + 6 × 3 = 37
61 = 37 + 6 × 4 = 61

Question 2.
How many small circles are there in each layer when the elders are stacked in the shape of a triangle? What number sequence does it form?
Solution:

Number of layers	1	2	3	4	5	6	7
Number of small circle	1	3	6	10	15	21	28

Sequence : 1, 3, 6, 10, 15, 21, 28, …………………..