In the binomial expansion of (a-b)^n, n ≥5, the sum of 5^th and 6^th terms is zero, then (a)/(

step 1 In this problem, we are given that the sum of the fifth and the sixth term in the expansion of a

In the binomial expansion of (a-b)^n, n ≥5, the sum of 5^th ...

In the binomial expansion of $(a-b)^{n}, n \geq 5$, the sum of $5^{\text {th }}$ and $6^{\text {th }}$ terms is zero, then $\frac{a}{b}$ equals
(A) $\frac{5}{n-4}$
(B) $\frac{6}{n-5}$
(C) $\frac{n-5}{6}$
(D) $\frac{n-4}{5}$

Key Concepts

Binomial Coefficients and Their Relationships

Binomial coefficients, denoted by C(n, k) or 'n choose k', are key in determining the weights of the terms in the expansion. Their properties and relationships, such as their ratio between consecutive terms, often simplify solving equations involving sums or differences of specific terms in the expansion.

Setting Up Algebraic Equations from Term Conditions

When given a condition involving specific terms of a binomial expansion (for instance, requiring the sum of two terms to be zero), one translates the condition into an algebraic equation involving binomial coefficients, powers of variables, and their signs. Solving this equation often involves comparing coefficients and deducing relationships between the variables, leading to the required result.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions raised to a power in the form (x + y)^n. It states that the expansion is a sum of terms involving binomial coefficients, which allows us to write any such power as a sum of individual terms. This concept is fundamental for problems requiring term identification and coefficient comparison in algebra.

General Term in a Binomial Expansion

In any binomial expansion, the general term is given by T_(k+1) = C(n, k)x^(n-k)y^k. When the binomial expression involves a subtraction, such as (a - b)^n, the general term becomes T_(k+1) = C(n, k)a^(n-k)(-b)^k, introducing alternating signs. Understanding this allows one to identify specific terms and set up equations based on given term relationships.

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