Question 1

Q1: Can you think of other examples where mathematics helps us in our everyday lives?

Accepted Answer

Mathematics is used in many aspects of our everyday lives. Here are some examples: 1.  Shopping: Calculating total bills, discounts, and taxes involves basic arithmetic. 2.  Cooking and Baking: Measuring ingredients accurately requires understanding fractions and ratios. 3.  Time Management: Reading...

Question 2

Q2: How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy a...

Accepted Answer

Mathematics has been fundamental to the advancement of human civilization in countless ways: 1.  Science and Technology: The laws of physics, chemistry, and biology are expressed in mathematical equations. This allows us to carry out scientific experiments, understand the universe, and develop new t...

Question 3

Q1: Can you recognise the pattern in each of the sequences in Table 1?

Accepted Answer

Yes, each sequence in Table 1 follows a specific rule or pattern:    1, 1, 1, 1, 1, 1, 1, ... (All 1's): Every number in the sequence is 1.    1, 2, 3, 4, 5, 6, 7, ... (Counting numbers): The sequence starts with 1, and each subsequent number is obtained by adding 1 to the previous number.    1, 3,...

Question 4

Q2: 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 wh...

Accepted Answer

Here are the sequences with their next three numbers and the rule for each:    (All 1's): 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...     Rule: Every number in the sequence is 1.    (Counting numbers): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...     Rule: Start with 1 and add 1 to the previous number to get the next n...

Question 5

Q1: Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!

Accepted Answer

The next picture for each sequence is described below:    All 1's: The next picture is a single dot, just like all the others.    Counting numbers: The next picture is a single row of 8 dots.    Odd numbers: The next picture is an L-shaped arrangement of 15 dots added around a $7 	imes 7$ square of...

Question 6

Q2: Why are $1,3,6,10,15, \ldots$ called triangular numbers? Why are $1,4,9,16,25, \ldots$ called square numbers or squares? Why are $1,8,27,64,125, \ldot...

Accepted Answer

These sequences are named after the geometric shapes their numbers can form:    Triangular numbers: The numbers in the sequence $1, 3, 6, 10, 15, \ldots$ can be arranged to form equilateral triangles. For example, 3 dots can form a triangle with 2 dots on each side, 6 dots can form a triangle with 3...

Question 7

Q3: 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...

Accepted Answer

To illustrate that 36 is both a square and a triangular number, you can draw two different arrangements of 36 dots: 1.  As a square: Draw a grid of dots with 6 rows and 6 columns. This arrangement will contain $6 	imes 6 = 36$ dots and will form a perfect square. 2.  As a triangle: Draw a triangle...

Question 8

Q4: What would you call the following sequence of numbers? 1, 7, 19, 37, 61, ... That's right, they are called hexagonal numbers! Draw these in your noteb...

Accepted Answer

To find the next number in the sequence: First, let's find the pattern by looking at the differences between consecutive terms:    $7 - 1 = 6$    $19 - 7 = 12$    $37 - 19 = 18$    $61 - 37 = 24$ The differences are multiples of 6: $6, 12, 18, 24$. The next difference in this pattern should be $30$....

Question 9

Q5: Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3?

Accepted Answer

Yes, there are several pictorial ways to visualize these sequences. Powers of 2 (1, 2, 4, 8, ...): One way is to use a branching diagram.  1.  Start with a single point (representing 1). 2.  From that point, draw two branches, leading to two new points (representing 2). 3.  From each of these two po...

Question 10

Q1: 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

Yes, a pictorial explanation can be found by looking at the diagonals of a square grid of dots. Consider a square grid of dots, for example, a $4 	imes 4$ square which has 16 dots. 1.  Start at one corner. There is 1 dot. 2.  The next diagonal line of dots parallel to the main diagonal has 2 dots....

Question 11

Q2: 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+\cdots+99+100+99+\cdots...

Accepted Answer

Pattern: The sum $1+2+\ldots+(n-1)+n+(n-1)+\ldots+2+1$ is equal to $n^2$. To Find: The value of $1+2+3+\cdots+99+100+99+\cdots+3+2+1$. Solution: In this sum, the peak number 'n' is 100. This sum represents the total number of dots in a $100 	imes 100$ square grid. Therefore, the value of the sum is...

Question 12

Q3: 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

1. Adding the All 1's sequence up:    1st term: $1$    2nd term: $1+1 = 2$    3rd term: $1+1+1 = 3$    4th term: $1+1+1+1 = 4$ The resulting sequence is 1, 2, 3, 4, ..., which is the Counting numbers sequence. 2. Adding the All 1's sequence up and down:    1st term: $1$    2nd term: $1+1+1 = 3$    3...

Question 13

Q4: Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?

Accepted Answer

1. The sequence: When we add the counting numbers up, we get:    1st term: $1$    2nd term: $1+2 = 3$    3rd term: $1+2+3 = 6$    4th term: $1+2+3+4 = 10$ The resulting sequence is 1, 3, 6, 10, ..., which is the Triangular numbers sequence. 2. Pictorial Explanation: We can visualize this by building...

Question 14

Q5: What happens when you add up pairs of consecutive triangular numbers? That is, take $1+3,3+6,6+10,10+15, \ldots$ Which sequence do you get? Why? Can y...

Accepted Answer

1. The sequence: Let's calculate the sums:    $1+3 = 4$    $3+6 = 9$    $6+10 = 16$    $10+15 = 25$ The resulting sequence is 4, 9, 16, 25, ..., which is the Square numbers sequence (starting from $2^2$). 2. Pictorial Explanation: A picture can explain why this happens. Take two consecutive triangul...

Question 15

Q6: 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...

Accepted Answer

1. Adding up powers of 2: Let's calculate the sums:    $1 = 1$    $1+2 = 3$    $1+2+4 = 7$    $1+2+4+8 = 15$ The resulting sequence is 1, 3, 7, 15, ... 2. Adding 1 to each number:    $1+1 = 2$    $3+1 = 4$    $7+1 = 8$    $15+1 = 16$ After adding 1, the new sequence is 2, 4, 8, 16, ..., which is the...

Question 16

Q7: What happens when you multiply the triangular numbers by 8 and add 1? Which sequence do you get? Can you explain it with a picture?

Accepted Answer

Correction: The provided text seems to have a typo in the solution key, which refers to multiplying by 6. The question asks to multiply by 8. Let's solve the question as written. 1. The sequence: The triangular numbers are 1, 3, 6, 10, 15, ... Let's multiply each by 8 and add 1:    $(1 	imes 8) + 1...

Question 17

Q8: 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 u...

Accepted Answer

1. The sequence: Let's calculate the sums:    $1$    $1+7 = 8$    $1+7+19 = 27$    $1+7+19+37 = 64$ The resulting sequence is 1, 8, 27, 64, ..., which is the Cube numbers sequence ($1^3, 2^3, 3^3, 4^3, \ldots$). 2. Explanation using a cube: This pattern relates to building larger cubes from smaller...

Question 18

Q9: 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?

Accepted Answer

Here is one example of a pattern: Pattern: The sum of the first 'n' counting numbers is equal to the nth triangular number. This is the definition of triangular numbers, but we can also relate it to other sequences. For example, the sum of a counting number and the next one is the difference between...

Question 19

Q1: By drawing a similar picture, can you say what is the sum of the first 10 odd numbers?

Accepted Answer

Pattern: The sum of the first 'n' odd numbers is equal to $n^2$. To Find: The sum of the first 10 odd numbers. Solution: Here, $n=10$. According to the pattern, the sum will be $10^2$. $$10^2 = 10 	imes 10 = 100$$ This corresponds to the total number of dots in a $10 	imes 10$ square grid. Final A...

Question 20

Q2: 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

Pattern: The sum of the first 'n' odd numbers is equal to $n^2$. To Find: The sum of the first 100 odd numbers. Solution: Here, $n=100$. According to the pattern, the sum will be $100^2$. $$100^2 = 100 	imes 100 = 10000$$ This corresponds to the total number of dots in a $100 	imes 100$ square gri...

NCERT Solutions