Problem 58 - Precalculus: Mathematics for Calculus

Question

Add each of the first five rows of Pascal's triangle, as indicated. Do you see a pattern? $$\begin{aligned}&\begin{array}{c}1+1=? \\\1+2+1=? \\\1+3+3+1=? \\\1+4+6+4+1=?\end{array}\\\&1+5+10+10+5+1=? \end{aligned}$$ On the basis of the pattern you have found, find the sum of the \(n\) th row: $$\left(\begin{array}{l}n \\\0\end{array}\right)+\left(\begin{array}{l}n \\\1\end{array}\right)+\left(\begin{array}{l} n \\\2\end{array}\right)+\cdots+\left(\begin{array}{l}n \\\n\end{array}\right)$$ Prove your result by expanding \((1+1)^{n}\) using the Binomial Theorem.

Step-by-Step Solution

Answer

The sum of the n-th row is \(2^n\).

Step 1: Add the elements of the first row

The first row of Pascal's triangle is \(1\). The sum is simply \(1\).

Step 2: Add the elements of the second row

The second row of Pascal's triangle is \(1, 1\). Add these together: \(1 + 1 = 2\).

Step 3: Add the elements of the third row

The third row is \(1, 2, 1\). Adding them gives \(1 + 2 + 1 = 4\).

Step 4: Add the elements of the fourth row

The fourth row is \(1, 3, 3, 1\). Their sum is \(1 + 3 + 3 + 1 = 8\).

Step 5: Add the elements of the fifth row

The fifth row is \(1, 4, 6, 4, 1\). The sum is \(1 + 4 + 6 + 4 + 1 = 16\).

Step 6: Add the elements of the sixth row

The sixth row is \(1, 5, 10, 10, 5, 1\). Compute their sum: \(1 + 5 + 10 + 10 + 5 + 1 = 32\).

Step 7: Identify the pattern

The sums of the rows are increasingly doubling from \(1\): \(1, 2, 4, 8, 16, 32\), which follows the pattern \(2^0, 2^1, 2^2, 2^3, 2^4, 2^5\). The pattern suggests that the sum of the elements in the \(n\)-th row is \(2^n\).

Step 8: Express the sum of the n-th row using the pattern

For the \(n\)-th row, the formula for the sum of its elements is \(2^n\).

Step 9: Prove using the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). For \(a = 1\) and \(b = 1\), we have \((1+1)^n = \sum_{k=0}^{n} \binom{n}{k} 1^{n-k} 1^k = \sum_{k=0}^{n} \binom{n}{k}\). Thus, the sum of the n-th row coefficients of Pascal's triangle is \((1+1)^n = 2^n\), confirming our pattern.

Key Concepts

Binomial Theorem

The Binomial Theorem is a key mathematical concept that allows us to expand expressions raised to a power. Specifically, it states that for any two numbers, \(a\) and \(b\), and any non-negative integer \(n\), the expansion of \((a + b)^n\) is a sum involving terms of the form \(\binom{n}{k} a^{n-k} b^k\). Here, \(\binom{n}{k}\) represents the binomial coefficient, which counts the number of ways to choose \(k\) elements from a set of \(n\) elements, and is expressed as \(\frac{n!}{k!(n-k)!}\).
The relevance of this theorem to Pascal's triangle is profound. When expanding \((1+1)^n\), each coefficient corresponds to an entry in the \(n\)-th row of Pascal's triangle. Therefore, the sum of the coefficients, which are the entries in a particular row, equals \(2^n\). This provides a neat way to calculate the sum of the elements in any row of Pascal's triangle, as demonstrated in the exercise.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting combinations and permutations, and has close links with both the Binomial Theorem and Pascal's Triangle. Each number in Pascal's Triangle, known as a binomial coefficient, corresponds to the number of combinations you can choose from a larger set. This is why combinatorics is central to understanding the triangle.
If you consider it, the Triangle is a visual representation of the various ways to choose subsets from a larger set. For example, the third row, \(1, 2, 1\), describes the number of ways to select 0, 1, or 2 elements from a two-element set. This understanding of combinatorics within Pascal's Triangle illustrates the power of counting and combinations, as we explore different patterns and possibilities in mathematics.

Patterns in Mathematics

Math is filled with fascinating patterns, and Pascal's Triangle is a classic example. By examining the exercise, we notice that the sums of the rows follow a clear doubling pattern: \(1, 2, 4, 8, 16, 32\), corresponding to \(2^0, 2^1, 2^2, 2^3, 2^4, 2^5\). Such patterns are not only pleasing to discover but also useful for predicting future behavior without calculation.
Patterns can serve as a shortcut or a method to delve deeper into mathematical relationships and ideology. Recognizing that the sum of each row in Pascal's Triangle follows a geometric progression of powers of two not only simplifies calculations but also allows us to draw connections to the Binomial Theorem, where each row represents the coefficients of the expanded form of \((1+1)^n\). Understanding these patterns helps students to grasp abstract concepts more intuitively by visualizing them.