Problem 11 - Algebra 2 and Trigonometry

Question

Write 10 lines of the Pascal Triangle, starting with 1

Step-by-Step Solution

Answer

The 10 lines are: 1; 1, 1; 1, 2, 1; 1, 3, 3, 1; 1, 4, 6, 4, 1; 1, 5, 10, 10, 5, 1; 1, 6, 15, 20, 15, 6, 1; 1, 7, 21, 35, 35, 21, 7, 1; 1, 8, 28, 56, 70, 56, 28, 8, 1; 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.

Step 1: Understanding the Pascal Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. We start with 1 at the top and proceed downwards with each row having one more element than the previous row.

Step 1: First Line

The first line of Pascal's Triangle is simply 1. This starts the triangle at the top.

Step 2: Second Line

The second line consists of two 1's. It is formed by adding the two numbers above it, but since there is only a single number (1), both numbers are 1.

Step 3: Third Line

For the third line, the numbers are derived by summing numbers from the line above. We have 1 from the start, 1+1=2 from the two 1's above (second line), and the line ends with 1: 1, 2, 1.

Step 4: Fourth Line

To form the fourth line, start with 1, then sum adjacent pairs from the line above: 1+2=3, 2+1=3, and end with 1: 1, 3, 3, 1.

Step 5: Fifth Line

The fifth line follows the same rule: 1 starts and ends the line, with 1+3=4, 3+3=6, 3+1=4 in between: 1, 4, 6, 4, 1.

Step 6: Sixth Line

The sixth line continues the pattern: starting and ending with 1, calculate intermediate values: 1+4=5, 4+6=10, 6+4=10, 4+1=5: 1, 5, 10, 10, 5, 1.

Step 7: Seventh Line

For the seventh line: 1 starts and ends, then 1+5=6, 5+10=15, 10+10=20, 10+5=15, 5+1=6: 1, 6, 15, 20, 15, 6, 1.

Step 8: Eighth Line

The eighth line is formed by: 1 starts and ends, then 1+6=7, 6+15=21, 15+20=35, 20+15=35, 15+6=21, 6+1=7: 1, 7, 21, 35, 35, 21, 7, 1.

Step 9: Ninth Line

In the ninth line: start and end with 1, then 1+7=8, 7+21=28, 21+35=56, 35+35=70, 35+21=56, 21+7=28, 7+1=8: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Step 10: Tenth Line

Finally, the tenth line begins and ends with 1, then follows: 1+8=9, 8+28=36, 28+56=84, 56+70=126, 70+56=126, 56+28=84, 28+8=36, 8+1=9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1.

Key Concepts

Binomial Coefficients

Binomial coefficients are a fascinating concept closely tied to Pascal's Triangle. They are used to express the coefficients in the expansion of a binomial expression \((a + b)^n\). Each number in the triangle represents a binomial coefficient.
For example, the numbers in the fifth row of Pascal's Triangle are 1, 4, 6, 4, 1. These correspond to the coefficients of \((a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\). You can see how using these coefficients simplifies expanding binomials significantly.
Let's understand how to find a specific binomial coefficient. If you are given the expression \((a + b)^n\) and wish to find the coefficient of \(a^k b^{n-k}\), simply look at Pascal's Triangle at row \(n\) (starting from zero) and column \(k\). This is precisely where the binomial coefficient is found.
Binomial coefficients are symmetrical; for a given row \(n\), the \(k\)-th coefficient from the start is equal to the \(n-k\)-th coefficient from the end. This symmetry is a beautiful feature that simplifies many calculations.

Combinatorics

Combinatorics is another mathematical field enriched by the use of Pascal's Triangle. Combinatorics deals with counting, arranging, and grouping objects. Binomial coefficients, as found in Pascal's Triangle, play a central role here.
Each binomial coefficient \(\binom{n}{k}\) represents the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order. This is a common problem in combinatorics.

• The formula for finding a binomial coefficient is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
• Here, \(n!\) ("n factorial") is the product of all positive integers up to \(n\).

Pascal's Triangle provides a quick way to find these coefficients, helping to solve complex counting problems with ease. For example, if you want to know the number of ways to choose 3 elements from a set of 5, look at the fifth row and third column (since rows start with 0) in Pascal's Triangle, which gives you 10.
This integration with combinatorics highlights why Pascal's Triangle is a valuable tool in mathematics.

Triangular Numbers

Triangular numbers are another interesting mathematical concept connected to Pascal's Triangle. These numbers represent a pattern of dots forming an equilateral triangle.
The sequence of triangular numbers starts with 1, 3, 6, 10, etc. These numbers are found in the third diagonal of Pascal's Triangle. Each triangular number is the cumulative total of the first \(n\) natural numbers.

• The formula for the \(n\)-th triangular number is: \[T_n = \frac{n(n + 1)}{2}\]
• For example, \(T_3\) is the third triangular number which is \(\frac{3 \times 4}{2} = 6\).

This connection between triangular numbers and Pascal's Triangle helps to visualize and calculate these numbers efficiently. Through recognizing this diagonal in Pascal's Triangle, students can better utilize it to solve practical mathematical problems involving triangular numbers.