91Ó°ÊÓ

The binomial expansion is a critical tool in algebra, allowing us to expand expressions that are raised to a power. When you see something like \((a + b)^n\), it's a signal to use the Binomial Theorem.
This theorem states that:\[(a + b)^n = \sum_{k=0}^{n} C(n,k) a^{(n-k)}b^k\].Here,

• \(C(n, k)\) denotes a binomial coefficient, which is calculated as \(\binom{n}{k}\).
• \(a^{(n-k)}\) means \(a\) raised to the power of \((n-k)\).
• \(b^k\) means \(b\) raised to the power of \(k\).

In our problem,

• \(a\) takes the value \(2\).
• \(b\) takes the value \(1\).

So, when applying the binomial expansion, we're using:\((2+1)^n\), which simplifies to \(3^n\).The expression on the left \(\sum_{k=0}^{n} 2^k C(n,k)\) represents exactly this expansion.

Combinatorics is the branch of mathematics dealing with combinations of objects. It helps us count how different items can be arranged or combined.
When we use the binomial coefficient \(C(n, k)\), we are using the heart of combinatorics,where:

• \(n\) represents the total number of items or steps.
• \(k\) represents the number of items to choose or steps to take.

The formula for a binomial coefficient is:\[C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}\].This tells us how many ways we can choose \(k\) items from a set of \(n\).
In the context of our exercise, these coefficients help us determine how each term in the expansion \((2 + 1)^n\) is formed and calculated.

Algebra is central to solving many mathematical problems.It provides the framework to express numbers and relationships through variables and symbols.Using algebra, we can manipulate equations to explore their properties.
In this exercise, algebra comes into play by employing the Binomial Theorem—a powerful algebraic tool.It tells us how to expand expressions like \((x + y)^n\) through a methodical approach.By knowing how each term relates to the combinatoric concept of binomial coefficients, we can solve the expression \(\sum_{k=0}^{n} 2^k C(n,k) = 3^n\).Through algebra, we demonstrate that replacing variables with specific numbers, in this case \(x = 2\) and \(y = 1\),results in an equation that holds true for any positive integer \(n\). This illustrates the power of algebra to abstract real numbers into problem-solving techniques.