Question

Use the Binomial Theorem to prove that $$3^{n}=\sum_{k=0}^{n} C(n, k) 2^{n-k}$$

Short answer

Use the Binomial Theorem on \((2+1)^n\).

Step-by-step solution

Understand the Binomial Theorem

The Binomial Theorem states that for any positive integer \( n \), \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\). We need to apply this theorem to prove the given expression.

Apply the Binomial Theorem

Rewrite \(3^n\) in a form that allows us to use the Binomial Theorem. Notice that \(3^n = (2+1)^n\). By applying the Binomial Theorem: \[(2+1)^n = \sum_{k=0}^{n} \binom{n}{k} 2^{n-k} 1^k.\] This can be rewritten as \(\sum_{k=0}^{n} \binom{n}{k} 2^{n-k}\) because \(1^k = 1\).

Conclude the Proof

Since \(3^n\) equals \((2+1)^n\), which, by the Binomial Theorem, equals \(\sum_{k=0}^{n} \binom{n}{k} 2^{n-k}\), the given equation is proven to be true.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and analyzing possible combinations in sets. In the context of the Binomial Theorem, combinatorics plays an essential role in determining the coefficients. These coefficients, known as binomial coefficients, are denoted as \( \binom{n}{k} \). They represent the number of ways to choose \( k \) elements from \( n \) elements, disregarding the order of selection. This is expressed mathematically as:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \),

where \( n! \) ("n factorial") is the product of all positive integers up to \( n \).
In the provided exercise, combinatorics helps us understand how the powers of 2 and 1 are combined in different terms of the expansion of \((2+1)^n\). Each term corresponds to a different way to choose powers of 2 and 1 while ensuring their exponents sum to \( n \). This insight is the backbone of connecting the expression \( 3^n \) to its binomial expansion.

Proof by Induction

Proof by Induction is a powerful mathematical principle used to prove statements for all natural numbers. It's akin to dominoes: proving the first statement (base case) helps prove the next, and so on.
To prove a statement by induction, follow these steps:

• **Base Case**: Verify the statement for the initial value, usually \( n=0 \) or \( n=1 \).
• **Inductive Step**: Assume the statement holds for an arbitrary natural number \( n \). Then, prove it holds for \( n+1 \).

In the context of the binomial theorem and expressions like \( 3^n = \sum_{k=0}^{n} \binom{n}{k} 2^{n-k} \), while the problem itself was solved using direct application of the Binomial Theorem, proof by induction could be employed to independently verify the equality for each \( n \). This method stresses how understanding a statement for one case can build a bridge to understanding all cases sequentially.

Discrete Mathematics

Discrete Mathematics is a field concerned with objects that take distinct, separate values. It plays a fundamental role in computer science and logic. It includes structures like graphs, integers, and in our interest, the binomial theorem.
In discrete mathematics, the Binomial Theorem is appreciated because it based heavily on the idea of discrete counting and finite processes. Each element in its expansion involves discrete variables—integers involved in combinations. For instance, when expanding \((2+1)^n\), each term requires evaluating combinatorial choices, like how many ways 2 and 1 can be multiplied across different terms.
Understanding discrete mathematics enriches grasping the binomial formula's usability in various applications where continuity is not natural or possible. From calculating probabilities in finite sample spaces to designing algorithms, discrete structures provide necessary clarity and insight into the finite worlds.