Problem 24 - Algebra 2

Question

ToVS Jamila is making a wall with building blocks. The top row has one block, the second row has three, the third has five, and so on. How many rows can she make with a set of 100 blocks?

Step-by-Step Solution

Answer

Jamila can make 10 rows with 100 blocks.

Step 1: Identify the Pattern

Observe the pattern in the number of blocks per row. The sequence of blocks per row is an arithmetic progression starting from 1, with a common difference of 2. This means the number of building blocks in each row are 1, 3, 5, 7, etc.

Step 2: Use the Sum Formula

The sum of the first n terms of an arithmetic sequence can be found using the formula: \( S_n = \frac{n}{2} (2a + (n-1)d) \), where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

Step 3: Apply the Formula

In this problem, \( a = 1 \) and \( d = 2 \). We need to find the largest \( n \) such that \( S_n \leq 100 \). Substituting these values, the sum becomes: \( S_n = \frac{n}{2} (2*1 + (n-1)*2) \) = \( \frac{n}{2} (2 + 2n - 2) = n^2 \).

Step 4: Solve the Inequality

We solve the inequality \( n^2 \leq 100 \) to find the maximum number of rows, \( n \). Taking the square root of both sides gives \( n \leq \sqrt{100} \) which simplifies to \( n \leq 10 \).

Step 5: Verify the Solution

Calculate the sum of the first 10 odd numbers: 1 + 3 + 5 + ... + 19. Using the formula for the sum of the first n odd numbers, \( S_n = n^2 \), we get \( S_{10} = 10^2 = 100 \). Thus, she can exactly build 10 rows.

Key Concepts

Arithmetic Progression

An arithmetic progression, or arithmetic sequence, is a series of numbers where each term increases or decreases by a constant difference. This difference between consecutive terms is known as the common difference.

• In the example of Jamila's wall, the sequence of building blocks starts with 1 block and increases by 2 blocks in each subsequent row: 1, 3, 5, 7, etc.
• This creates a clear pattern of increasing the number of building blocks by the same amount each time, a hallmark of arithmetic progressions.
• The first term (often represented as \( a \)) in our sequence is 1, while the common difference \( d \) is 2.

Understanding an arithmetic progression helps us calculate the total number of items over several terms, such as building blocks in rows, easily and systematically.

Sum Formula for Arithmetic Sequence

The sum of an arithmetic sequence can be calculated using a specific formula. This allows us to find the total number of items without having to individually add each term manually.

• The sum \( S_n \) of the first \( n \) terms can be determined with the formula: \( S_n = \frac{n}{2} (2a + (n-1)d) \).
• In the case of Jamila's wall, \( a = 1 \) and \( d = 2 \). Therefore, applying these values, the formula simplifies to \( S_n = n^2 \).

With this formula, you're able to quickly find the sum of any number of terms in the sequence, streamlining problem-solving and making complex calculations more accessible.

Inequality Solving

Inequality solving involves finding a range of values that satisfy a given condition and is crucial in determining limits and capabilities.

• For Jamila’s wall, we need to find the maximum number of rows \( n \) that allows her not to exceed 100 building blocks. This translates to solving the inequality \( n^2 \leq 100 \).
• Taking the square root on both sides offers the solution \( n \leq 10 \), meaning she can have up to 10 rows without exceeding her block limit.

Inequality solving simplifies challenging scenarios into manageable equations, facilitating easier problem-solving strategies.

Pattern Recognition in Sequences

Recognizing patterns in sequences is a skill that can significantly enhance problem-solving abilities by identifying recurring themes or behaviors within a sequence.

• In the problem, the pattern of adding two more blocks per row (1, 3, 5, 7, ...) reveals an arithmetic progression.
• Observing such patterns assists in predicting future terms, identifying the sequence type, and applying the correct formulas for calculations.

Developing pattern recognition skills allows for efficient and accurate problem resolution, helping interpret complex information in a straightforward manner.