Problem 7 - A Gentle Introduction to the Art of Mathematics

Question

Prove the binomial theorem. $$ \forall n \in \mathbb{N}, \forall x, y \in \mathbb{R},(x+y)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) x^{n-k} y^{k} $$

Step-by-Step Solution

Answer

By using mathematical induction, it is proven that \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k}\) for any natural number \(n\) and any real numbers \(x\) and \(y\).

Step 1 - Understand the Statement

To prove the binomial theorem, we need to show that for any natural number \(n\) and any real numbers \(x\) and \(y\), the expansion of \((x+y)^n\) equals the sum of \( \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \)

Step 2 - Use Mathematical Induction

We'll use induction on \(n\). First, verify the base case where \(n=0\): \( (x+y)^0 = 1 \) and \( \sum_{k=0}^{0} \binom{0}{k} x^{0-k} y^{k} = \binom{0}{0} x^0 y^0 = 1 \). Base case holds true.

Step 3 - Inductive Step

Assume that the theorem holds for some integer \(n=k\), i.e., \( (x+y)^k = \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} \). Now show it holds for \(n=k+1\).

Step 4 - Expand \( (x+y)^{k+1} \)

Consider \( (x+y)^{k+1} = (x+y)(x+y)^k \). Using the induction hypothesis, substitute the expansion for \( (x+y)^k \): \( (x+y)^{k+1} = (x+y) \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} \)

Step 5 - Distribute

Apply distributive property: \( (x+y) \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} = x \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} + y \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} \)

Step 6 - Simplify

Rewrite the expression: \( x \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} + y \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j} \=\sum_{j=0}^{k} \binom{k}{j} x^{(k+1)-j} y^{j} + \sum_{j=0}^{k} \binom{k}{j} x^{k-j} y^{j+1} \)

Step 7 - Adjust Indices

Adjust the indices to combine the sums. Let \(i=j\) in the first sum and \(i=j+1\) in the second sum, then combine:\( \sum_{i=0}^{k} \binom{k}{i} x^{(k+1)-i} y^{i} + \sum_{i=1}^{k+1} \binom{k}{i-1} x^{(k+1)-(i-1)} y^{i} \). Now combine:\( x^{k+1} + \sum_{i=1}^{k} \left( \binom{k}{i} + \binom{k}{i-1} \right) x^{k+1-i} y^{i} + y^{k+1} \)

Step 8 - Binomial Coefficient Property

Use the property of binomial coefficients: \( \binom{k}{i} + \binom{k}{i-1} = \binom{k+1}{i} \). Thus,\( x^{k+1} + \sum_{i=1}^{k} \binom{k+1}{i} x^{k+1-i} y^{i} + y^{k+1} = \sum_{i=0}^{k+1} \binom{k+1}{i} x^{(k+1)-i} y^{i} \)

Step 9 - Conclusion

Since we have shown \( (x+y)^{k+1} = \sum_{i=0}^{k+1} \binom{k+1}{i} x^{(k+1)-i} y^{i} \), the inductive step is complete, and the binomial theorem holds for all natural numbers \(n\).

Key Concepts

Mathematical Induction

Mathematical induction is a powerful technique used to prove statements for all natural numbers. It involves two main steps:

• Base Case: Show that the statement is true for the initial value (usually n=0 or n=1).
• Inductive Step: Assume the statement is true for an arbitrary natural number k, then prove it's true for k+1.

By establishing these two steps, we can conclude that the statement is true for all natural numbers.

Binomial Coefficients

Binomial coefficients are key in the binomial theorem. They're represented as \( \binom{n}{k} \) and read as 'n choose k'. This represents the number of ways to choose k elements from a set of n elements, given by the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, ! denotes factorial, which means multiplying a series of descending natural numbers. Binomial coefficients appear in the binomial expansion of \((x + y)^n\), indicating how many ways each term can be formed by choosing powers of x and y.

Distributive Property

The distributive property is fundamental in algebra, and it's used extensively in proofs like the binomial theorem. It states that multiplying a sum by a number gives the same result as multiplying each addend individually and then summing up the products:
\[ a(b + c) = ab + ac \]
This property allows us to distribute terms over addition, simplifying complex expressions into manageable parts.

Natural Numbers

Natural numbers, denoted as \( \mathbb{N} \), are the set of positive integers starting from 1 (or 0 in some definitions). These numbers are fundamental in mathematics, particularly in the binomial theorem, which is proved for all natural numbers. They play a critical role in sequences, series, and many algebraic structures.