Rational Numbers | IT

Question 23

What patterns do you notice in the table above?

Solution

Understand the Question

A perfect square is obtained by multiplying an integer by itself ( $n^2 = n \times n$ ).
Examining consecutive perfect squares reveals two key patterns:
1. The sum of the first $n$ consecutive odd natural numbers starting from $1$ is equal to $n^2$ .
2. Each subsequent square $(n+1)^2$ can be obtained by adding the next consecutive odd number $(2n+1)$ to the preceding square $n^2$ .

Step 1 · List First Few Perfect Squares

A perfect square is formed by multiplying an integer by itself: Diagram 1

\begin{aligned} 1^2 &= 1 \times 1 = 1 \\ 2^2 &= 2 \times 2 = 4 \\ 3^2 &= 3 \times 3 = 9 \\ 4^2 &= 4 \times 4 = 16 \\ 5^2 &= 5 \times 5 = 25 \end{aligned}

Step 2 · Pattern 1: Sum of Consecutive Odd Numbers

Every square number can be expressed as the sum of consecutive odd numbers starting from $1$ :

\begin{aligned} 1^2 &= 1 \\ 2^2 &= 1 + 3 = 4 \\ 3^2 &= 1 + 3 + 5 = 9 \\ 4^2 &= 1 + 3 + 5 + 7 = 16 \\ 5^2 &= 1 + 3 + 5 + 7 + 9 = 25 \end{aligned}

In general, the sum of the first $n$ consecutive odd numbers is $n^2$ .

Step 3 · Pattern 2: Building Squares from the Previous Square

Each subsequent square is obtained by adding the next consecutive odd number to the preceding square:

$2^2 = 1^2 + 3 = 1 + 3 = 4$

$3^2 = 2^2 + 5 = 4 + 5 = 9$

$4^2 = 3^2 + 7 = 9 + 7 = 16$

$5^2 = 4^2 + 9 = 16 + 9 = 25$

In general, the next square is given by: $(n+1)^2 = n^2 + (2n+1)$

Answer

Sum of odd numbers: The sum of the first $n$ consecutive odd numbers starting from $1$ is equal to $n^2$ .

Successive squares: Each square is formed by adding the next consecutive odd number $(2n+1)$ to the previous square: $(n+1)^2 = n^2 + (2n+1)$ .

Common Mistakes

Skipping 1 in Odd Sums: The sum property only holds when starting from $1$ (e.g., $3 + 5 = 8$ , which is not a square).
Incorrect Odd Addition: Adding an arbitrary odd number rather than the exact next odd number $(2n+1)$ to get the next square.

More questions in IT

Context: Queen Ratnamanjuri left a puzzle in her will for her son Khoisnam and 99 relatives. They are in a room with 100 lockers, numbered 1 to 100.

Person 1 opens every locker.
Person 2 toggles every 2nd locker (closes if open, opens if closed).
Person 3 toggles every 3rd locker (3rd, 6th, 9th, $\dots$ ).
Person 4 toggles every 4th locker (4th, 8th, 12th, $\dots$ ). This continues until all 100 get their turn.

Q. Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer?

Hint: Find out how many times each locker is toggled.

Does every number have an even number of factors?

Can you use this insight to find more numbers with an odd number of factors?

Context: In a room with 100 lockers, only lockers whose numbers are square numbers remain open.

Q. Write the locker numbers that remain open.

Find the squares of the first 30 natural numbers and fill in the table below.

Context: Patterns and Properties of Perfect Squares

Find the squares of the first 30 natural numbers and fill in the table below.

Q. What patterns do you notice? Share your observations and make conjectures.

If a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?

Write 5 numbers such that you can determine by looking at their units digit that they are not squares.

Let us consider square numbers ending in 6: $16 = 4^2$ , $36 = 6^2$ , $196 = 14^2$ , $256 = 16^2$ , $576 = 24^2$ , and $676 = 26^2$ . Which of the following numbers have the digit 6 in the units place?

(i) $38^2$ (ii) $34^2$ (iii) $46^2$ (iv) $56^2$ (v) $74^2$ (vi) $82^2$

Q10

Find more such patterns by observing the numbers and their squares from the table you filled earlier.

Q11

If a number contains 3 zeros at the end, how many zeros will its square have at the end?

Q12

What do you notice about the number of zeros at the end of a number and the number of zeros at the end of its square? Will this always happen? Can we say that squares can only have an even number of zeros at the end?

Q13

What can you say about the parity of a number and its square?

Q14

Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?

Q15

How many square numbers are there between 1 and 100? How many are between 101 and 200? Using the table of squares you filled earlier, enter the values below, tabulating the number of squares in each block of 100. What is the largest square less than 1000?

Q16

Can you see any relation between triangular numbers and square numbers? Extend the pattern shown and draw the next term.

Q17

Find whether $1156$ and $2800$ are perfect squares using prime factorisation.

Q18

How many cubes of side $1\text{ cm}$ make a cube of side $2\text{ cm}$ ?

Q19

How many cubes of side $1\text{ cm}$ will make a cube of side $3\text{ cm}$ ?

Q20

Is 9 a cube?

Q21

Can you estimate the number of unit cubes in a cube with an edge length of 4 units?

Q22

Complete the table below.

Q23

What patterns do you notice in the table above?

Q24

We know that 0, 1, 4, 5, 6, 9 are the only last digits possible for squares. What are the possible last digits of cubes?

Q25

Similar to squares, can you find the number of cubes with 1 digit, 2 digits, and 3 digits? What do you observe?

Q26

Can a cube end with exactly two zeroes (00)? Explain.

Q27

The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.

Q28

Context: Look at the following pattern of consecutive odd numbers:

\begin{aligned} 1 &= 1 = 1^3 \\ 3 + 5 &= 8 = 2^3 \\ 7 + 9 + 11 &= 27 = 3^3 \\ 13 + 15 + 17 + 19 &= 64 = 4^3 \\ 21 + 23 + 25 + 27 + 29 &= 125 = 5^3 \\ 31 + 33 + 35 + 37 + 39 + 41 &= 216 = 6^3 \end{aligned}

Later in this series, we get the following set of consecutive numbers:

$91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109$

Q. Can you tell what this sum is without doing the calculation?

Q29

Find the cube roots of these numbers:

(i) $\sqrt[3]{64} =$

(ii) $\sqrt[3]{512} =$

(iii) $\sqrt[3]{729} =$

Q30

Compute successive differences over levels for perfect cubes until all the differences at a level are the same. What do you notice?