Problem 61 - Algebra and Trigonometry

Question

Describe the pattern in the exponents on \(b\) in the expansion of \((a+b)^{n}\)

Step-by-Step Solution

Answer

In the expansion of \((a+b)^{n}\), the exponent of \(b\) starts at 0 and increases by 1 for each term, up until \(n\) for the last term.

Step 1: Understand the binomial theorem

The binomial theorem provides us with an easy way to calculate the expansion of a binomial expression raised to a certain power. The theorem is written as: \((a+b)^{n} = a^n + ^nC_1 a^{n-1}b + ^nC_2 a^{n-2}b^2 + .... + ^nC_{n-1} a b^{n-1} + b^n\), where ^nC_k is the number of combinations of \(n\) items taken \(k\) at a time.

Step 2: Understand the pattern in the exponents

In each term of the expansion provided by the binomial theorem, the sum of the exponents of \(a\) and \(b\) is always \(n\). The power on \(b\) starts from 0 in the first term, then increases by 1 as we move right along the terms, and ends at \(n\) for the last term, which is where \(b\) is raised to the power \(n\). Simultaneously, the power on \(a\) starts from \(n\) and decreases by 1 as we move right along the terms, reaching 0 for the last term.

Step 3: Describe the pattern

The pattern in the exponents of \(b\) in the expansion of \((a+b)^{n}\) is that the power of \(b\) starts at 0 and increases by 1 for each successive term, ending at \(n\) for the last term in the expansion.

Key Concepts

Exponents pattern in binomial expansion

In the expansion of \((a+b)^n\) using the binomial theorem, the pattern of exponents on the terms can be easily observed. The binomial expansion consists of a sequence of terms where each term is a product of powers of \(a\) and \(b\).
For any term in the expansion, the sum of the exponents of \(a\) and \(b\) always equals \(n\).

• The power of \(b\) in each term starts at 0 and progresses upward by 1 in successive terms.
• Simultaneously, the power of \(a\) starts at \(n\) and decreases by 1 in each subsequent term.

This systematic and predictable increase of the exponent of \(b\) from 0 up to \(n\) contributes to a neat and structured arrangement of terms within the polynomial expansion.

Combinatorial coefficients

Combinatorial coefficients, often referred to as binomial coefficients, play a crucial role in the binomial theorem. The coefficients in the expanded form \((a+b)^n\) are expressed using the symbol \(^nC_k\), which represents the number of ways to choose \(k\) items from \(n\) without regard to order. These coefficients are calculated using the formula:\[^nC_k = \frac{n!}{k!(n-k)!}\]

• Each term's coefficient is derived from binomial coefficients, making them pivotal in determining the contribution of each term in a binomial expansion.
• Understanding these coefficients helps in determining the "weight" or influence of each term in the expansion based on its placement.

The distribution of these coefficients creates a symmetrical pattern, enhancing the expansion's harmony and balance.

Polynomial expansion

The process of expanding \((a+b)^n\) to form a polynomial showcases the power of polynomial expansion and binomial theorem together. Binomial expansion transforms a simple expression into a sum of multiple terms. Each term in the polynomial is a distinct and individually calculated expression that contributes to the overall sum.

• The expanded polynomial typically contains \(n+1\) terms.
• Each of these terms consists of a coefficient, and variables \(a\) and \(b\) raised to particular powers following the rules of the binomial theorem.

Polynomial expansion provides a methodical approach to breaking down powers of binomials, thus simplifying complex calculations.

Pascal's triangle

Pascal's triangle is a geometric arrangement of numbers forming an equilateral triangle, where each number is the sum of the two numbers directly above it in the previous row. This insightful tool is invaluable when dealing with binomial expansions as its rows correspond to the coefficients in the expansion of \((a+b)^n\).

• The \(n\)-th row in Pascal’s Triangle gives the binomial coefficients for \((a+b)^n\).
• This helps in quickly finding the coefficients needed for any term in the expansion without having to compute combinations manually.

Using Pascal's triangle not only facilitates easy computation but also vividly illustrates the inherent symmetry and balance within binomial expansions, making it an essential tool in understanding polynomial behavior.