The Leaning Tower of Cards. You are attempting to break the world record for building the tallest freestanding house of cards. As of January 1, 1998, the record stood at 68 stories, reaching a height of 3.9 meters using standard playing cards. To beat the record, you plan to build a 69-story house of cards. A single- story house of cards requires 3 cards. A double-story house (2-storey house) requires 8 cards. A triple-story house (3-storey house) needs 15 cards. Find a general formula or a mathematical model for the nth term. How many cards do you need to build a 69-story cards and break the world record?

To find the general formula for the number of cards needed to build an \(n\)-story house of cards, we start by analyzing the pattern given in the problem:

- 1-story house: 3 cards
- 2-story house: 8 cards
- 3-story house: 15 cards

We observe that the difference between consecutive terms forms a pattern:
- \(8 - 3 = 5\)
- \(15 - 8 = 7\)

These differences increase by 2 each time, suggesting that the formula is quadratic.

### Step 1: General form of a quadratic formula
The general form of a quadratic formula for the number of cards needed for an \(n\)-story house of cards is:
\[
a n^2 + b n + c
\]
We need to determine the constants \(a\), \(b\), and \(c\).

### Step 2: Create equations using known terms
We know the number of cards for \(n = 1\), \(n = 2\), and \(n = 3\):

1. For \(n = 1\): 
\[
a(1)^2 + b(1) + c = 3
\]
\[
a + b + c = 3 \quad \text{(Equation 1)}
\]

2. For \(n = 2\): 
\[
a(2)^2 + b(2) + c = 8
\]
\[
4a + 2b + c = 8 \quad \text{(Equation 2)}
\]

3. For \(n = 3\): 
\[
a(3)^2 + b(3) + c = 15
\]
\[
9a + 3b + c = 15 \quad \text{(Equation 3)}
\]

### Step 3: Solve the system of equations
Now we solve the system of three equations:

1. \(a + b + c = 3\)
2. \(4a + 2b + c = 8\)
3. \(9a + 3b + c = 15\)

#### Subtract Equation 1 from Equation 2:
\[
(4a + 2b + c) - (a + b + c) = 8 - 3
\]
\[
3a + b = 5 \quad \text{(Equation 4)}
\]

#### Subtract Equation 2 from Equation 3:
\[
(9a + 3b + c) - (4a + 2b + c) = 15 - 8
\]
\[
5a + b = 7 \quad \text{(Equation 5)}
\]

#### Subtract Equation 4 from Equation 5:
\[
(5a + b) - (3a + b) = 7 - 5
\]
\[
2a = 2
\]
\[
a = 1
\]

#### Substitute \(a = 1\) into Equation 4:
\[
3(1) + b = 5
\]
\[
3 + b = 5
\]
\[
b = 2
\]

#### Substitute \(a = 1\) and \(b = 2\) into Equation 1:
\[
1 + 2 + c = 3
\]
\[
c = 0
\]

### Step 4: General formula
Thus, the general formula for the number of cards required for an \(n\)-story house of cards is:
\[
n^2 + 2n
\]

### Step 5: Calculate for 69 stories
To calculate the number of cards needed for a 69-story house of cards, substitute \(n = 69\) into the formula:
\[
69^2 + 2(69) = 4761 + 138 = 4899
\]

So, you need **4899 cards** to build a 69-story house of cards and break the world record.

---

Would you like further details on how this formula works, or any other questions? Here are some follow-up questions:

1. How does the quadratic pattern explain the increasing number of cards?
2. How many cards are needed for a 100-story house of cards?
3. Could you derive an alternate formula for larger structures?
4. How does the card structure physically support itself as stories increase?
5. Can you model the weight distribution for the tallest house of cards?

**Tip**: When analyzing patterns in sequences, always check if the differences between terms suggest a quadratic, cubic, or higher degree formula!

The Leaning Tower of Cards. You are attempting to break the world record for building the tallest freestanding house of cards. As of January 1, 1998, the record stood at 68 stories, reaching a height of 3.9 meters using standard playing cards. To beat the record, you plan to build a 69-story house of cards. A single-story house of cards requires 3 cards. A double-story house (2-storey house) requires 8 cards. A triple-story house (3-storey house) needs 15 cards. Find a general formula or a mathematical model for the nth term. How many cards do you need to build a 69-story cards and break the world record?

Discover how to calculate the number of cards needed for a 69-story house of cards to break the world record. This problem uses quadratic equations to model the growth in the number of cards as stories increase. Suitable for students in Grades 9-12, this example covers pattern recognition and algebraic concepts.

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