Question 4

The sum of the first few odd numbers always makes a square number.

Step 1 — See the pattern

Look at the picture given. The first odd number is 1. It forms a square of $1 \times 1$. $1 = 1^2$

Now add the next odd number, which is 3. We have $1 + 3$ dots. This makes a square of $2 \times 2$. $1 + 3 = 4$

$\boxed{2^2}$

Next, add the odd number 5. We have $1 + 3 + 5$ dots. This makes a square of $3 \times 3$. $1 + 3 + 5 = 9$

$\boxed{3^2}$

Let us add the odd number 7. We have $1 + 3 + 5 + 7$ dots. This makes a square of $4 \times 4$. $1 + 3 + 5 + 7 = 16$

$\boxed{4^2}$

Step 2 — Find the rule

We see a clear pattern here. The sum of the first 1 odd number is $1^2$. The sum of the first 2 odd numbers is $2^2$. The sum of the first 3 odd numbers is $3^2$. The sum of the first 4 odd numbers is $4^2$. So, the sum of the first 'n' odd numbers is $n^2$.

Step 3 — Calculate the sum

We need the sum of the first 100 odd numbers. Using our rule, this sum will be $100^2$. $100^2 = 100 \times 100$

$\boxed{10,000}$

Answer

(i) The sum of the first 100 odd numbers is 10,000.