Question

a) Penelope claims that if you read any row in Pascal's triangle as a single number, it can be expressed in the form \(11^{m},\) where \(m\) is a whole number. Do you agree? Explain. b) What could \(m\) represent?

Short answer

Penelope's claim is true for the first few rows of Pascal's Triangle. Here, m represents the row number.

Step-by-step solution

- Understand Pascal’s Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. The first few rows of Pascal's Triangle are as follows: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 and so on.

- Express Rows as Numbers

Read each row as a single number:- First row: 1- Second row: 11- Third row: 121- Fourth row: 1331- Fifth row: 14641

- Express Numbers in the Form of Powers of 11

Check if these numbers can be expressed as 11^0 = 1 11^1 = 11 11^2 = 121 11^3 = 1331 11^4 = 14641 Notice that for the initial rows of Pascal’s triangle, the numbers can indeed be expressed as powers of 11.

- Evaluate Larger Rows

For larger rows, the numbers won't follow the exact pattern: The sixth row is 1 5 10 10 5 1, which gives 15101051 when read as a single number. Clearly, this is not a power of 11.

- Conclude if Penelope’s Claim is Correct

Penelope's claim is only partially correct. Reading the rows of Pascal’s triangle as single numbers matches the form 11^m for the initial few rows, but not for higher rows.

- Define m in context

In context, m could represent the row number in Pascal’s Triangle. For small values, the row number as the exponent gives the correct power of 11. However, this relationship does not hold as numbers grow larger.

Key concepts

binomial coefficients

Binomial coefficients are the numbers that appear in Pascal's Triangle. They are crucial in algebra and combinatorics. Each entry in Pascal's Triangle represents a binomial coefficient, which can be calculated using the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). These coefficients are essential for expanding binomial expressions and understanding distributions in probability and statistics. Studying Pascal's Triangle helps in grasping how binomial coefficients build upon each other.

patterns in mathematics

Pascal's Triangle is a goldmine of patterns. By examining the structure closely, you can see symmetrical patterns and fractals develop. For instance, each row is symmetrical, meaning it reads the same forwards and backwards. Additionally, the sums of numbers along each diagonal create interesting sequences. Observing these patterns teaches you how mathematical elements interconnect. Recognizing such patterns is invaluable for problem-solving and developing analytical thinking skills.

powers of numbers

In the initial rows of Pascal's Triangle, you might notice a resemblance to powers of 11. For example, \(11^0 = 1\), \(11^1 = 11\), \(11^2 = 121\), and so on. However, as you move to higher rows, the entries become more complex, and this pattern does not hold. This observation highlights an intriguing interplay between binomial coefficients and powers of numbers, showcasing how mathematical properties evolve. Understanding this relationship underscores the depth and beauty of mathematical structures.

mathematical properties

Pascal's Triangle isn't just a collection of numbers; it embodies various mathematical properties. These include combinatorial identities, such as the hockey-stick identity and the Binomial Theorem. Each row also corresponds to the coefficients in the expansion of a binomial expression \((a + b)^n\). By studying these properties, students can gain insight into more complex mathematical concepts and apply them in real-world scenarios. Embracing these properties fosters a deeper appreciation and comprehension of mathematics.