To solve the problem, we need to determine the pattern in the number of dots across the figures and then apply this pattern to figure out how many dots will be in the 22nd figure.

1. Identifying the number of dots:
   • Figure 1 has 5 dots.
   • Figure 2 has 13 dots.
   • Figure 3 has 25 dots.
   • Figure 4 has 41 dots.

We can observe that the number of dots increases in each figure following a specific pattern. Let's examine the differences between consecutive figures:

• From Figure 1 to Figure 2: $13 - 5 = 8$
• From Figure 2 to Figure 3: $25 - 13 = 12$
• From Figure 3 to Figure 4: $41 - 25 = 16$

The difference between successive figures increases by 4, which suggests that the pattern follows a quadratic sequence.

2. Finding the quadratic formula: We know that the number of dots follows a quadratic sequence, which can be expressed as: $a_n = an^2 + bn + c$ where $a_n$ is the number of dots in the $n$-th figure.

Using the known values:

• $a_1 = 5$
• $a_2 = 13$
• $a_3 = 25$

We substitute these values into the quadratic formula to set up a system of equations:

$a(1)^2 + b(1) + c = 5$ $a(2)^2 + b(2) + c = 13$ $a(3)^2 + b(3) + c = 25$

This simplifies to: $a + b + c = 5 \quad (1)$ $4a + 2b + c = 13 \quad (2)$ $9a + 3b + c = 25 \quad (3)$

Solving this system of equations will give us the values of $a$, $b$, and $c$. Let's solve for these values:

• Subtract (1) from (2): $(4a + 2b + c) - (a + b + c) = 13 - 5$ gives $3a + b = 8$.
• Subtract (2) from (3): $(9a + 3b + c) - (4a + 2b + c) = 25 - 13$ gives $5a + b = 12$.

Now we solve the system:

• Subtract the first equation from the second: $(5a + b) - (3a + b) = 12 - 8$ gives $2a = 4$, so $a = 2$.
• Substitute $a = 2$ into $3a + b = 8$: $3(2) + b = 8$, so $b = 2$.

Substitute $a = 2$ and $b = 2$ into $a + b + c = 5$: $2 + 2 + c = 5$, so $c = 1$.

Thus, the quadratic formula for the number of dots is: $a_n = 2n^2 + 2n + 1$

3. Finding the number of dots in the 22nd figure: Now, substitute $n = 22$ into the formula: $a_{22} = 2(22)^2 + 2(22) + 1 = 2(484) + 44 + 1 = 968 + 44 + 1 = 1013$

Therefore, the 22nd figure will contain 1013 dots.

Would you like more details or have any questions about the solution?

Additional related questions:

1. What type of sequence is the pattern of dots, and how can it be identified?
2. How can we derive a quadratic equation from any set of three numbers?
3. How does the difference method help in identifying patterns in sequences?
4. Can we use an alternative method like recursion to find the number of dots?
5. What would the number of dots be in the 30th figure using the same pattern?

Tip:

Quadratic sequences often have constant second differences between terms, making it easier to recognize and solve them.