Question 1

Q.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?

Accepted Answer

These sequences are named based on the shapes formed by arranging dots:
- Triangular numbers (1, 3, 6, 10, 15, ...) represent dots arranged in an equilateral triangle.
- Square numbers (1, 4, 9, 16, 25, ...) represent dots arranged in a perfect square.
- Cube numbers (1, 8, 27, 64, 125, ...) represent dots arranged in a perfect cube.
Refer to Table 2 on page 4 for pictorial illustrations. — Triangular numbers count dots that can form an equilateral triangle; square numbers count dots forming a square; cube numbers count dots forming a cube. Visualizing these arrangements helps understand the naming.

Question 2

Q.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!

Accepted Answer

36 is a triangular number because it can be arranged as a triangle with 8 rows (sum of first 8 natural numbers = 36).
It is also a square number because 6 × 6 = 36, so dots can be arranged in a 6 by 6 square.
Draw these arrangements in your notebook to visualize. — Triangular number for 36: 1+2+3+4+5+6+7+8 = 36.
Square number for 36: 6×6 = 36.
Drawing these helps understand the dual nature of 36.

Question 3

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

Accepted Answer

The answer given is 61.
(Note: The original question seems incomplete or missing the sequence itself in the text.) — The question as given is incomplete in the source text; only the answer '61' is provided.

Question 4

Q.5. Can you think of pictorial ways to visualise the sequence of powers of 2? powers of 3?

Accepted Answer

For powers of 2, refer to the sequence given on page 6.
For powers of 3, one way is to visualize cubes growing in size, as shown in the images (img-12.jpeg to img-15.jpeg) illustrating the cubes of 1, 3, 9, 27, etc. — Pictorial visualization helps understand exponential growth. Powers of 2 can be shown as doubling squares or dots; powers of 3 can be shown as cubes increasing in volume.

Question 5

- By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?

Accepted Answer

The sum of the first 10 odd numbers is 100.
This can be visualized by arranging dots in a square of side 10, since the sum of first n odd numbers equals n². — Sum of first n odd numbers = n².
For n=10, sum = 10² = 100.

Question 6

Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers?

Accepted Answer

The sum of the first 100 odd numbers is 10,000.
Because sum of first n odd numbers = n², for n=100, sum = 100² = 10,000. — Sum of first n odd numbers = n².
For n=100, sum = 100² = 10,000.

Question 7

Q.1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e.,  $1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, \ldots$ , gives square numbers?

Accepted Answer

One way is to visualize the sequence as layers of dots added symmetrically up and down forming perfect squares.
For example:
- 1 = 1²
- 1 + 2 + 1 = 4 = 2²
- 1 + 2 + 3 + 2 + 1 = 9 = 3²
This pattern continues, showing the sums form square numbers. — Adding counting numbers up and down creates symmetric layers that form squares.
This is shown pictorially in the figure (img-16.jpeg).

Question 8

Q.2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be the value of  $1 + 2 + 3 + \ldots + 99 + 100 + 99 + \ldots + 3 + 2 + 1$ ?

Accepted Answer

The sum is 10,000.
Because the sequence adds up from 1 to 100 and back down to 1, forming a perfect square of side 100.
Hence, sum = 100² = 10,000. — Sum of 1 to 100 is 5050.
Adding 99 down to 1 is 4950.
Total sum = 5050 + 4950 = 10,000.

Question 9

Q.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?

Accepted Answer

Adding the All 1's sequence up gives:
1 + (1 + 1) + (1 + 1 + 1) + (1 + 1 + 1 + 1) + ...
which is the sequence: 1, 2, 3, 4, ... (Counting numbers).
Adding the All 1's sequence up and down gives:
1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, ...
which forms the square numbers sequence: 1, 4, 9, 16, ... — Summing 1's repeatedly counts natural numbers.
Summing up and down creates symmetric sums forming squares.

Question 10

Q.4. Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?

Accepted Answer

Adding counting numbers up gives the triangular number sequence:
1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, ...
which equals 1, 3, 6, 10, ...
A pictorial representation is given in Table 2 on page 4.
Try also for isosceles right triangles. — Triangular numbers represent dots arranged in triangles.
Sum of first n natural numbers = n(n+1)/2.

Question 11

Q.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?

Accepted Answer

Adding pairs of consecutive triangular numbers gives:
1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, ...
which is the sequence of square numbers.
This happens because two consecutive triangular numbers combine to form a perfect square.
Refer to Table 2 on page 4 for pictorial explanation. — Triangular numbers represent half of a square; adding two consecutive triangular numbers forms a full square.

Question 12

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

Accepted Answer

Adding powers of 2 starting with 1 gives the sequence:
1, 3, 7, 15, 31, ...
Adding 1 to each number gives:
2, 4, 8, 16, 32, ...
This happens because sum of first n powers of 2 is 2^n - 1.
Adding 1 results in powers of 2.
Refer to the picture on page 6. — Sum of powers of 2: $1 + 2 + 4 + \dots + 2^{n-1} = 2^n - 1$.
Adding 1 yields $2^n$, the next power of 2.

Question 13

Q.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?

Accepted Answer

Multiplying triangular numbers by 6 and adding 1 gives:
(1×6)+1=7, (3×6)+1=19, (6×6)+1=37, (10×6)+1=61, (15×6)+1=91, ...
This sequence is: 7, 19, 37, 61, 91, ...
Refer to the picture in Q.4 on page 5 for explanation. — This sequence relates to hexagonal numbers and their properties.
Visualizing the pattern helps understand the sequence.

Question 14

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

Accepted Answer

Adding hexagonal numbers gives the sequence:
1, 8, 27, 64, ...
which are cube numbers.
Refer to Table 2 on page 4 for pictorial explanation. — Sum of hexagonal numbers corresponds to cubes.
Visualizing with cubes helps understand the sequence.

Question 15

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

Accepted Answer

Yes, the patterns are:
- 3, 4, 5, 6, 7, 8, 9, 10: Number of sides of regular polygons.
- 1, 3, 6, 10, 15: Triangular numbers.
- 1, 4, 9, 16, 25: Square numbers.
- 3, 3×4, 3×4×4, 3×4×4×4, 3×4×4×4×4: Sequence involving powers of 4 multiplied by 3. — Each sequence corresponds to geometric or arithmetic patterns related to shapes and their properties.

Patterns In

Patterns In — Study Notes

1.1 What is Mathematics?

1.2 Patterns in Numbers

1.3 Visualising Number Sequences

Practice Questions — Patterns In

All 10 Chapters in Ganita Prakash