This pattern works because the sum of consecutive odd numbers can be visualized as building up a square. Start with a 1 × 1 square (1). To get a 2 × 2 square (4), you add 3 units in an 'L' shape around the 1 × 1 square. To get a 3 × 3 square (9), you add 5 units in an 'L' shape around the 2 × 2 square. Each time, you add the next odd number to form the next larger square.

This pattern works because the sum of consecutive odd numbers can be visualized as building up a square. Start with a 1 × 1 square (1). To get a 2 × 2 square (4), you add 3 units in an 'L' shape around the 1 × 1 square. To get a 3 × 3 square (9), you add 5 units in an 'L' shape around the 2 × 2 square. Each time, you add the next odd number to form the next larger square.

Observe the pattern: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9. Using this pattern, find the sum of the first 8 odd numbers and explain why the pattern works.

Answer: The sum of the first 8 odd numbers is 64. The pattern works because the sum of consecutive odd numbers forms perfect squares, which can be visualized geometrically or proven using the arithmetic progression sum formula.

Step-by-step solution

Step 1: Analyze the given pattern

We are given a pattern where the sum of the first $n$ odd numbers is equal to $n^2$ . Let's observe the first few terms to confirm this. The first odd number is 1, and its sum is $1 = 1^2$ . The sum of the first two odd numbers ( $1+3$ ) is $4 = 2^2$ . The sum of the first three odd numbers ( $1+3+5$ ) is $9 = 3^2$ .

Step 2: Generalize the pattern

From the observed pattern, we can generalize that the sum of the first $n$ odd numbers is equal to $n$ squared. This is a known property of arithmetic progressions.

Step 3: Calculate the sum of the first 8 odd numbers

Using the generalized pattern, to find the sum of the first 8 odd numbers, we simply need to calculate $8^2$ . This gives us $64$ .

Step 4: Explain why the pattern works (Visual Proof)

This pattern works because the sum of consecutive odd numbers can be visualized as building up a square. Start with a $1 \times 1$ square (1). To get a $2 \times 2$ square (4), you add 3 units in an 'L' shape around the $1 \times 1$ square. To get a $3 \times 3$ square (9), you add 5 units in an 'L' shape around the $2 \times 2$ square. Each time, you add the next odd number to form the next larger square.

Step 5: Explain why the pattern works (Arithmetic Progression)

The sequence of odd numbers ( $1, 3, 5, \dots$ ) is an arithmetic progression with the first term $a=1$ and common difference $d=2$ . The sum of the first $n$ terms of an arithmetic progression is given by the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ . Substituting $a=1$ and $d=2$ into the formula, we get $S_n = \frac{n}{2}[2(1) + (n-1)2] = \frac{n}{2}[2 + 2n - 2] = \frac{n}{2}[2n] = n^2$ . This proves the pattern mathematically.

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