Question 12 Marks
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Question 22 Marks
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?
Question 32 Marks
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?
Question 42 Marks
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?
Question 52 Marks
Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?
Answer
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Question 62 Marks
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?
Question 72 Marks
Answer
View full question & answer→Yes
• Power of 2 We can visualize the powers of 2 as squares where each subsequent square has twice the number of smaller squares as the previous one.
• Power of 3 We can visualize the powers of 3 as cubes, where each subsequent cube has three times the number of smaller cubes as the previous one.
• Power of 2 We can visualize the powers of 2 as squares where each subsequent square has twice the number of smaller squares as the previous one.
• Power of 3 We can visualize the powers of 3 as cubes, where each subsequent cube has three times the number of smaller cubes as the previous one.
Question 82 Marks
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 is the sequence?
That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number is the sequence?
Question 92 Marks
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!
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!
Question 102 Marks
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?
Answer
View full question & answer→1, 3, 6, 10, 15, ... sequence forms a triangles, therefore it is called triangular numbers.
1, 4, 9, 16, 25, ... sequence forms a square, therefore, it is called square numbers.
1. 8. 27, 64, 125, ... sequence forms a cube, therefore, it is called cubes.
1, 4, 9, 16, 25, ... sequence forms a square, therefore, it is called square numbers.
1. 8. 27, 64, 125, ... sequence forms a cube, therefore, it is called cubes.
Question 112 Marks
Copy the pictorial representations of the number sequences in Table 2 in your notebook and draw the next picture for each sequence.
Question 122 Marks
Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequences! After each sequences, write in your own words what is the rule for forming the numbers in the sequence.
Answer
View full question & answer→• All 1's 1, 1, 1, 1, ... (The next three numbers are 1, 1, 1. The rule is that every number is 1.)
• Counting numbers 1, 2, 3, 4, 5, 6, 7, ... (The next three numbers are 8, 9, 10. The rule is to add 1 to the previous number.)
• Odd numbers 1, 3, 5, 7, 9, 1,... (The next three numbers are 15, 17, 19. The rs to add 2 to the previous number.)n numbers 2, 4, 6, 8, 10, 12, 14, ... (The next three is to add 2 to the previous number.)
• Triangular numbers 1, 3, 6, 10, 15, 21, 28, ... (The next three numbers are 36, 45, 55. The rule is to add the next natural number is sequence.)
• Squares 1, 4, 9, 16, 25, 36, 49, ... (The next three numbers are 64, 81, 100. The rule is to square the next natural number.)
• Cubes 1, 8, 27, 64, 125, 216, ... (The next three numbers are 343, 512, 729. The rule is to cube the next natural number.)
• Virahanka numbers 1, 2, 3, 5, 8, 13, 21, ... (The next three numbers are 34, 55, 89. The rule is to add the last two numbers.)
• Powers of 2 1, 2, 4, 8, 16, 32, 64, ... (The next three numbers are 128, 256, 512. The rule is to multiply the last number by 2.)
• Powers of 3 1, 3, 9, 27, 81, 243, 729, ... (The next three numbers are 2187, 6561, 19683. The rule is to multiply the last number by 3.)
• Counting numbers 1, 2, 3, 4, 5, 6, 7, ... (The next three numbers are 8, 9, 10. The rule is to add 1 to the previous number.)
• Odd numbers 1, 3, 5, 7, 9, 1,... (The next three numbers are 15, 17, 19. The rs to add 2 to the previous number.)n numbers 2, 4, 6, 8, 10, 12, 14, ... (The next three is to add 2 to the previous number.)
• Triangular numbers 1, 3, 6, 10, 15, 21, 28, ... (The next three numbers are 36, 45, 55. The rule is to add the next natural number is sequence.)
• Squares 1, 4, 9, 16, 25, 36, 49, ... (The next three numbers are 64, 81, 100. The rule is to square the next natural number.)
• Cubes 1, 8, 27, 64, 125, 216, ... (The next three numbers are 343, 512, 729. The rule is to cube the next natural number.)
• Virahanka numbers 1, 2, 3, 5, 8, 13, 21, ... (The next three numbers are 34, 55, 89. The rule is to add the last two numbers.)
• Powers of 2 1, 2, 4, 8, 16, 32, 64, ... (The next three numbers are 128, 256, 512. The rule is to multiply the last number by 2.)
• Powers of 3 1, 3, 9, 27, 81, 243, 729, ... (The next three numbers are 2187, 6561, 19683. The rule is to multiply the last number by 3.)
Question 132 Marks
Can you recognise the pattern in each of the sequence in Table 1?
Answer
View full question & answer→Yes, we can recognise the pattern in each of the sequence in table 1.
Question 142 Marks
To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ with 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)
Answer
View full question & answer→There are 3, 12, 48, 192, 768 line segments in each shape of the Koch Snowflake.
The corresponding number sequence is 3, 12, 48, ....
i.e. 3 times the powers of 4.
The corresponding number sequence is 3, 12, 48, ....
i.e. 3 times the powers of 4.
Question 152 Marks
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?)
Answer
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Question 162 Marks
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?
Answer
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Question 172 Marks
Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
Answer
View full question & answer→There are 1, 3, 6, 10, 15 lines respectively in each shape in the sequence of complete graphs.
We get a sequence of triangular numbers.
We get a sequence of triangular numbers.
Question 182 Marks
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?
Answer
Sequence - 3, 4, 5, 6, 7, 8, 9, 10
We get, a sequence of counting numbers
Also, the number of corners in the shapes is same as the number of sides.
View full question & answer→| Shape | Number of sides | Number of corners |
| Triangle | 3 | 3 |
| Quadrilateral (Square) | 4 | 4 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 6 |
| Heptagon | 7 | 7 |
| Octagon | 8 | 8 |
| Nonagon | 9 | 9 |
| Decagon | 10 | 10 |
We get, a sequence of counting numbers
Also, the number of corners in the shapes is same as the number of sides.