https://file.aiquickdraw.com/mathbot/file/17279721266744wz5dndn.jpg

This image depicts a pattern where shapes made of dots are arranged in a triangular pattern. The question asks for the number of dots in the 51st shape in this sequence. Let's break it down:

- The 1st shape has 5 dots.
- The 2nd shape has 13 dots.
- The 3rd shape has 25 dots.

The number of dots seems to be increasing in a pattern based on the shape's position. Let's analyze the sequence to find the rule.

### Step 1: Check the differences between the numbers of dots
- 2nd shape: \( 13 - 5 = 8 \)
- 3rd shape: \( 25 - 13 = 12 \)

The differences are increasing by 4. This suggests the sequence follows a quadratic pattern.

### Step 2: Use the quadratic formula
We can assume the number of dots \( a_n \) follows a quadratic equation of the form:

\[
a_n = An^2 + Bn + C
\]

Using the first three terms:
- \( a_1 = 5 \)
- \( a_2 = 13 \)
- \( a_3 = 25 \)

We can create a system of equations:
1. \( A(1)^2 + B(1) + C = 5 \)
2. \( A(2)^2 + B(2) + C = 13 \)
3. \( A(3)^2 + B(3) + C = 25 \)

Simplifying:
1. \( A + B + C = 5 \)
2. \( 4A + 2B + C = 13 \)
3. \( 9A + 3B + C = 25 \)

Now, solving this system will give us the values of \( A \), \( B \), and \( C \).

Let me calculate these values for you.The solution to the quadratic equation is:

\[
A = 2, \quad B = 2, \quad C = 1
\]

Thus, the number of dots \( a_n \) in the \( n \)-th shape follows the formula:

\[
a_n = 2n^2 + 2n + 1
\]

### Step 3: Calculate the number of dots in the 51st shape
We can now substitute \( n = 51 \) into the formula:

\[
a_{51} = 2(51)^2 + 2(51) + 1
\]

Let me calculate that for you.The 51st shape will have **5,305 dots**.

Would you like further details or have any questions about this? Here are some related questions you might find useful:

1. How does the quadratic sequence relate to triangular number patterns?
2. Can you derive a different formula for similar dot patterns?
3. What happens if the difference between dots in consecutive shapes changes differently?
4. How does the sequence behave for much larger values of \(n\)?
5. How can we use this approach for non-quadratic sequences?

**Tip:** Quadratic sequences often appear in patterns where the second difference between consecutive terms is constant!

Learn how to calculate the number of dots in the 51st shape using quadratic sequences. The pattern follows a formula derived from analyzing the differences in consecutive shapes. Suitable for students in Grades 9-11.

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