Question

a) Determine the sum of the numbers in each of the first five rows in Pascal's triangle. b) What is an expression for the sum of the numbers in the ninth row of Pascal's triangle? c) What is a formula for the sum of the numbers in the \(n\) th row?

Short answer

a) 1, 2, 4, 8, 16. b) \(2^8 = 256\). c) \(2^n\).

Step-by-step solution

Understanding Pascal's Triangle

Pascal's Triangle is a triangular array of binomial coefficients. Each row corresponds to the coefficients of the binomial expansion \( (a + b)^n \). The rows start with index 0.

Sum of Numbers in First Row

First row (index 0): \(1\). Sum = 1.

Sum of Numbers in Second Row

Second row (index 1): \(1, 1\). Sum = 1 + 1 = 2.

Sum of Numbers in Third Row

Third row (index 2): \(1, 2, 1\). Sum = 1 + 2 + 1 = 4.

Sum of Numbers in Fourth Row

Fourth row (index 3): \(1, 3, 3, 1\). Sum = 1 + 3 + 3 + 1 = 8.

Sum of Numbers in Fifth Row

Fifth row (index 4): \(1, 4, 6, 4, 1\). Sum = 1 + 4 + 6 + 4 + 1 = 16.

Summarizing Sums of First Five Rows

First five sums are: \[ 1, 2, 4, 8, 16 \].

Recognizing the Pattern

The sum of the numbers in the \(n\)th row of Pascal's Triangle is \(2^n\).

Expression for Ninth Row

For the ninth row (index 8), \ n = 8 \. The sum is \( 2^8 = 256 \).

General Formula for Sum of Numbers in the n-th Row

General formula for the sum of the numbers in the \(n\)th row of Pascal's Triangle is \( 2^n \).

Key concepts

Binomial Coefficients

To understand Pascal's Triangle, we first need to grasp the concept of binomial coefficients. A binomial coefficient appears in the expansion of a binomial raised to a power, such as \((a + b)^n\). Each coefficient is represented as \({{n} \choose {k}}\), which reads as 'n choose k'. This counts the number of ways to choose k elements from a set of n elements without regard to the order of selection.
In Pascal's Triangle:

• The top row represents \((a+b)^0\) and only contains the binomial coefficient \(1\).
• The second row represents \((a+b)^1\), showing the coefficients 1 and 1. Every subsequent row lists all the coefficients from the expansion of \((a+b)^n\), with the row numbers starting at 0.
  Each element in the triangle is the sum of the two elements directly above it. This recurrence relation helps to quickly build Pascal's Triangle.
  

Row Sum Formula

One interesting feature of Pascal's Triangle is the sum of the numbers in each row. By observing the sums of the first few rows, you can see an emerging pattern:

• Row 0: Sum is 1.
• Row 1: Sum is 2 (1 + 1).
• Row 2: Sum is 4 (1 + 2 + 1).
• Row 3: Sum is 8 (1 + 3 + 3 + 1).
• Row 4: Sum is 16 (1 + 4 + 6 + 4 + 1).

You may notice that these sums are powers of 2: \(2^0, 2^1, 2^2, 2^3, 2^4\).This tells us an important rule: The sum of the numbers in the nth row of Pascal's Triangle is given by \(2^n\). For example, sum of numbers in the 9th row (where \(n=8\)) is \(2^8=256\).
So, if you want to find the sum of any row's numbers in Pascal's Triangle, just use \(2^n\), where n is the row number.

Mathematical Patterns

Pascal's Triangle is not only useful for calculating binomial coefficients and row sums, but also for discovering interesting mathematical patterns. For example:

• **Symmetry**: Each row is symmetric. For example, Row 4: \(1, 4, 6, 4, 1\).
• **Fibonacci Sequence**: By summing up the shallow diagonals of Pascal's Triangle, you get the Fibonacci numbers.Start from the edge and look at the numbers leaning diagonally: 1, 1, 2, 3, 5, 8, etc.
• **Pascal's Identity**: Each number is the sum of the two numbers directly above it: \({{n-1} \choose {k-1}} + {{n-1} \choose {k}} = {{n} \choose {k}}\).

These patterns and identities make Pascal's Triangle a practical tool in combinatorial mathematics, algebra, and even computer science. Recognizing these underlying patterns helps us appreciate the interconnectedness of these mathematical concepts.