Question 1

Adding up consecutive odd numbers always gives us a square number.

Step 1 — Finding the pattern

Let us look at the numbers. We add odd numbers one by one. The first odd number is 1. The sum is 1. This is $1 \times 1$.

The first two odd numbers are 1 and 3. We add them together. The sum is 4. This is $2 \times 2$.

The first three odd numbers are 1, 3, and 5. We add them together. The sum is 9. This is $3 \times 3$.

The first four odd numbers are 1, 3, 5, 7. We add them together. The sum is 16. This is $4 \times 4$.

We see a clear pattern here. The sum of $N$ odd numbers is $N \times N$. We can write this as $N^2$.

$\boxed{\text{Sum of } N \text{ odd numbers} = N^2}$

Step 2 — Why this happens

Let us think about squares. A square has equal sides. We can draw squares using small blocks. Look at the diagram below.

The first odd number is 1. It makes a $1 \times 1$ square. It uses 1 block.

The next odd number is 3. We add 3 blocks around the first square. This makes a $2 \times 2$ square. It uses $1 + 3 = 4$ blocks.

The next odd number is 5. We add 5 blocks around the $2 \times 2$ square. This makes a $3 \times 3$ square. It uses $4 + 5 = 9$ blocks.

The next odd number is 7. We add 7 blocks around the $3 \times 3$ square. This makes a $4 \times 4$ square. It uses $9 + 7 = 16$ blocks.

Each time, we add the next odd number. This new number always fits perfectly. It forms the next bigger square. The number of blocks added is always odd. We add blocks along two sides. We also add one corner block. So, the total is always odd. So, the sum of $N$ odd numbers is always $N^2$.

Step 3 — Will it happen forever?

Yes, this pattern will happen forever. We can always make a bigger square. We just add the next odd number of blocks. Let us have an $N \times N$ square. We add $2N + 1$ blocks. This $2N + 1$ is always the next odd number. It will form a $(N+1) \times (N+1)$ square. So, this pattern never ends. It is a basic property of numbers.

Answer

(i) Each odd number added perfectly completes the next square. (ii) Yes, this pattern will happen forever. (iii) We always add the next odd number to form a new square.