We can observe that the number of dots in each pattern seems to follow a certain pattern itself. Let's analyze the given sequence:

• 1st pattern: 5 dots
• 2nd pattern: 12 dots
• 3rd pattern: 21 dots

To find how the number of dots changes, let's look at the differences between consecutive terms:

$12 - 5 = 7$ $21 - 12 = 9$

It appears that the differences are increasing by 2. This suggests that the sequence might be quadratic in nature. We can assume the number of dots follows a quadratic formula of the form:

$n = an^2 + bn + c$

Where $n$ is the pattern number, and $a$, $b$, and $c$ are constants we need to determine. Using the given data points, we can set up the following system of equations:

1. For the first pattern ($n = 1$, $dots = 5$):
   $a(1)^2 + b(1) + c = 5 \quad \Rightarrow \quad a + b + c = 5$
2. For the second pattern ($n = 2$, $dots = 12$):
   $a(2)^2 + b(2) + c = 12 \quad \Rightarrow \quad 4a + 2b + c = 12$
3. For the third pattern ($n = 3$, $dots = 21$):
   $a(3)^2 + b(3) + c = 21 \quad \Rightarrow \quad 9a + 3b + c = 21$

Now, solving this system of equations:

1. $a + b + c = 5$
2. $4a + 2b + c = 12$
3. $9a + 3b + c = 21$

Subtracting the first equation from the second:

$(4a + 2b + c) - (a + b + c) = 12 - 5$ $3a + b = 7 \quad \Rightarrow \quad b = 7 - 3a$

Next, subtract the second equation from the third:

$(9a + 3b + c) - (4a + 2b + c) = 21 - 12$ $5a + b = 9 \quad \Rightarrow \quad b = 9 - 5a$

Now, equating the two expressions for $b$:

$7 - 3a = 9 - 5a$ $2a = 2 \quad \Rightarrow \quad a = 1$

Substitute $a = 1$ into $b = 7 - 3a$:

$b = 7 - 3(1) = 4$

Now substitute $a = 1$ and $b = 4$ into the first equation:

$1 + 4 + c = 5 \quad \Rightarrow \quad c = 0$

Thus, the quadratic formula for the number of dots is:

$n = n^2 + 4n$

To find the number of dots in the 10th pattern, substitute $n = 10$ into the formula:

$n = 10^2 + 4(10) = 100 + 40 = 140$

So, the 10th pattern has 140 dots.

Would you like further details on the process, or do you have other questions?

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Related Questions:

1. How would the sequence change if the second pattern had 15 dots instead of 12?
2. Can you generalize this method to find a formula for any quadratic sequence?
3. How can we verify if a sequence is quadratic using differences?
4. What is the significance of the constant term $c$ in the quadratic formula?
5. How would the result change if the differences between consecutive dots increased by a different number?

Tip: When dealing with sequences where differences between terms increase steadily, always consider checking for a quadratic relationship by calculating second differences!