Hasif uses white and black sticks to form figures that follow a pattern as shown below. (Figures 1 to 5 are shown.)

1. Determine the number of white sticks for Figure 6

Observe the pattern for the number of white sticks:

• Figure 1: 2
• Figure 2: 2
• Figure 3: 8
• Figure 4: 8
• Figure 5: 18

The pattern repeats each value twice. The sequence of unique values is $2, 8, 18, \dots$, which can be expressed as $2 \times 1^2, 2 \times 2^2, 2 \times 3^2, \dots$.

• For Figures 1 and 2: $2 \times 1^2 = 2$
• For Figures 3 and 4: $2 \times 2^2 = 8$
• For Figures 5 and 6: $2 \times 3^2 = 18$

Thus, the number of white sticks for Figure 6 is 18.

2. Determine the number of black sticks for Figure 6

Observe the pattern for the number of black sticks:

• Figure 1: 0
• Figure 2: 4
• Figure 3: 4
• Figure 4: 12
• Figure 5: 12

The pattern repeats each value twice starting from Figure 2. The sequence of unique non-zero values is $4, 12, 24, \dots$, which can be expressed as $4 \times 1$, $4 \times 3$, $4 \times 6$, where the multipliers are triangular numbers $1, 3, 6, \dots$.
Alternatively, we can find the total sticks first ($T_n = n(n+1)$) and subtract white sticks:

• Total sticks for Figure 6: $6 \times 7 = 42$
• Black sticks for Figure 6: $42 - 18 = 24$.

3. Determine the total number of sticks for Figure 6

Observe the pattern for the total number of sticks:

• Figure 1: $1 \times 2 = 2$
• Figure 2: $2 \times 3 = 6$
• Figure 3: $3 \times 4 = 12$
• Figure 4: $4 \times 5 = 20$
• Figure 5: $5 \times 6 = 30$

Following the pattern $n(n+1)$, for Figure 6:
Total sticks = $6 \times 7 = 42$.

1. Identify the algebraic rule for the total number of sticks based on the table

Observe the total number of sticks for each figure:

• Figure 1: $2 = 1 \times 2$
• Figure 2: $6 = 2 \times 3$
• Figure 3: $12 = 3 \times 4$
• Figure 4: $20 = 4 \times 5$
• Figure 5: $30 = 5 \times 6$

In general, the total number of sticks for Figure $n$ is given by the formula:
$\text{Total} = n(n + 1)$

2. Set up and solve the equation to find the figure number $n$ for 650 sticks

Set the formula equal to 650:
$n(n + 1) = 650$
$n^2 + n - 650 = 0$

Since $n$ is a positive integer, we can find two consecutive numbers that multiply to 650. Let's estimate using square roots:
$\sqrt{650} \approx 25.5$

Check $25 \times 26$:
$25 \times 26 = 650$

Thus, $n = 25$.