Problem 60 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

In the expansion of \((a+b)^{n}\), the exponent on \(a\) starts at \(n\) and decreases by 1 for each subsequent term, until it reaches 0.

Step 1: Description of the Binomial Theorem

The binomial theorem (also known as binomial expansion) describes the algebraic expansion of powers of a binomial. According to this theorem, it can be written as: \((a+b)^{n}=∑_{k=0}^{n}( )^{n}C_{k} * a^{n-k}*b^{k}\), where \(( )^{n}C_{k}\) represents the number of combinations of n items taken k at the time.

Step 2: Pattern of the Exponents on a

Observing the expression \((a+b)^{n}=∑_{k=0}^{n}( )^{n}C_{k} * a^{n-k}*b^{k}\), it can be seen that the exponent of \(a\) starts at \(n\) (when \(k=0\)) and decreases by 1 each time until it reaches 0 (when \(k=n\)). The sum of the exponents of \(a\) and \(b\) in each term always adds up to \(n\).

Step 3: Explaination of the Pattern

The pattern holds because each term of the binomial expression represents the number of ways to pick certain items out of a larger set, mirroring the way the exponents decrease for \(a\) and increase for \(b\) throughout the expansion. For instance, when expanding a binomial expression to the power of \(n\), in each term of the expansion, \(a\) is chosen \(n-k\) times and \(b\) is chosen \(k\) times.

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a fundamental concept in algebra that involves expressing the power of a binomial, which is an expression made up of two terms added together, such as \(a + b\), as a sum of simpler terms. This expansion follows a precise pattern described by the binomial theorem. When we expand \(a+b\)^n\, where \(n\) is a non-negative integer, the result is a sum of terms in which the exponents of \(a\) and \(b\) have specific patterns and are accompanied by coefficients known as binomial coefficients.

These coefficients can be determined using the formula \(^{n}C_{k}\) which represents the number of combinations, and corresponds to the coefficient in front of the term where \(a\) is raised to the power of \(n-k\) and \(b\) is raised to the power of \(k\). For example, in the expansion of \(a+b\)^3\, the terms would be \(a^3 + 3a^2b + 3ab^2 + b^3\). The coefficients 1, 3, 3, and 1 correspond to the number of ways to choose 0, 1, 2, and 3 instances of \(b\), respectively, from the three possible slots.

Understanding binomial expansion is crucial for solving a wide range of problems in algebra, from simplifying expressions to calculating probabilities in statistics.

Patterns in Exponents

When dealing with the expansion of binomials, the pattern of exponents plays a pivotal role in determining the individual terms. In the binomial theorem, for the expansion of \(a+b\)^n\, the exponent on \(a\) starts at \(n\) and decreases by 1 with each subsequent term, while the exponent on \(b\) starts at 0 and increases by one until it too reaches \(n\).

Illustration of the Exponent Pattern

Consider the term in the expansion of \(a+b\)^5\ where \(k = 2\). According to the pattern, \(a\) will have an exponent of \(5-2=3\) while \(b\) will have an exponent of 2. This term would therefore be \(10a^3b^2\), with 10 being the binomial coefficient calculated as \(^{5}C_{2}\).

The sum of the exponents in each term equals the overall power \(n\), demonstrating the fundamental principle that the expansion of a binomial is all about the different ways to distribute the exponents between the two variables within the constraints of the equation \(a+b\)^n.

Mastering Combinations

The concept of combinations is integral to understanding the coefficients found in binomial expansion. The binomial coefficient \(^{n}C_{k}\) in the expression of \(a+b\)^n\ tells us how many different ways we can select \(k\) items from a larger set of \(n\) without regard to the order of selection.

Significance of Combinations in Binomial Expansion

When expanding a binomial expression, combinations indicate the number of ways that the variables \(a\) and \(b\) can be paired together. As an example, in \(a+b\)^4\, the term with \(a^2b^2\) has a coefficient of \(^{4}C_{2}=6\), reflecting that there are 6 unique combinations of two \(a\)s and two \(b\)s that can be formed when selecting from a total of four elements (since we're considering the term where \(k = 2\)).

Comprehending the calculation and meaning of combinations not only aids in binomial expansion but also in fields like probability and combinatorics where the arrangement and selection of objects are fundamental. By mastering this concept, students can approach a wide array of mathematical problems with confidence and clarity.