Problem 31 - Discrete Mathematics with Applications

Question

Using the binomial theorem, prove each. \(\sum_{r=0}^{n} 2^{r}\left(\begin{array}{l}n \\ r\end{array}\right)=3^{n}\)

Step-by-Step Solution

Answer

By using the binomial theorem with appropriate values for a and b (a=1 and b=2), we can expand \((1+2)^n = 3^n\), which simplifies to \(\sum_{r=0}^{n} \binom{n}{r} 2^r = 3^n\). This matches the expression we needed to prove, confirming that \(\sum_{r=0}^{n} 2^{r}\binom{n}{r} = 3^{n}\).

Step 1: Identify the appropriate values for a and b

As we want to prove the given equation using the binomial theorem, let's go for a = 1 and b = 2, so that it has the power of 2 which we need for our equation. Now that we have the appropriate values for a and b, let's apply the binomial theorem:

Step 2: Apply the binomial theorem

Using the binomial theorem, we have: \((a + b)^n = (1 + 2)^n = 3^n\) Now, expanding the left hand side using the binomial theorem: \((1+2)^n\ =\ \sum_{r=0}^{n} \binom{n}{r} 1^{n-r}2^{r}\) But the powers of 1 have no effect on the sum, so we can simplify the expression:

Step 3: Simplify the expression

Now, we can simplify the equation as: \[\sum_{r=0}^{n} \binom{n}{r} 2^r\ =\ 3^n\] This matches the expression we needed to prove, so now we can conclude that our proof is done.

Step 4: Conclusion

By using the binomial theorem with appropriate values for a and b, we have proven that \(\sum_{r=0}^{n} 2^{r}\binom{n}{r} = 3^{n}\).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that focuses on counting, arrangement, and combination of objects. It's incredibly useful when solving problems that involve discrete and finite elements. In the context of the binomial theorem, combinatorics is the key to understanding how different values are selected and combined.
One of the most vital concepts in combinatorics is the binomial coefficient, denoted as \( \binom{n}{r} \). This represents the number of ways to choose \( r \) elements from a set of \( n \) elements, regardless of the order. It's expressed as a fraction:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \(!\) is the factorial function.

This coefficient is crucial in expanding binomial expressions. For instance, if we want to expand \((a + b)^n\), the binomial theorem shows us that it can be represented as a sum of terms, each including a binomial coefficient. Combinatorics helps us understand how these coefficients arise and how they contribute to the formula, making it possible to solve seemingly complex problems with relatively simple mathematical expressions.

Mathematical Proof

Mathematical proofs are the backbone of ensuring statements in mathematics are true. Proofs bring rigor and logic, providing a solid foundation for understanding complex concepts. Here, we focus on proving identities using the binomial theorem.
In the provided exercise, mathematical proof is used to validate the equation:

• \( \sum_{r=0}^{n} 2^{r} \binom{n}{r} = 3^{n} \)

To accomplish this, we first identify the correct values for variables \( a \) and \( b \) in the binomial expression \((a + b)^n\). By choosing \( a = 1 \) and \( b = 2 \), we align our expression with the desired form:

• \((1 + 2)^n = 3^n\)
• Expanded as \((1+2)^n\ =\ \sum_{r=0}^{n} \binom{n}{r} 1^{n-r}2^{r}\)

After simplifying (because 1 raised to any power is always 1), this matches the target equation:

• \( \sum_{r=0}^{n} \binom{n}{r} 2^r\ =\ 3^n\)

Hence, mathematical proof serves to demonstrate the validity of the expression, ensuring the math holds under scrutiny.

Discrete Mathematics

Discrete mathematics is the study of structures that are distinct and separate (as opposed to continuous). It covers topics like graphs, integers, and yes, combinatorics. The finite nature of discrete mathematics makes it essential for computer science, algorithms, and encryption.
In this exercise, we're dealing with discrete quantities—integer values and fixed powers. The binomial theorem is a discrete mathematical tool that helps us expand powers of binomials into sums involving discrete amounts. For example:

• The finite series \( \sum_{r=0}^{n} 2^{r}\) involves fixed powers and an integer coefficient \( \binom{n}{r} \).
• The expression \(3^n\) represents a distinct, singular quantity.

Thus, we're working entirely within the realm of discrete numbers. In discrete mathematics, proving identities like the one here illustrates how finite sums and combinatorial coefficients combine to create meaningful relationships and solutions that are key to problem-solving in countless mathematical applications.