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

step 1 in this problem we are given a minus b power n and we are given that n is greater than equal to

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

In the binomial expansion of $(a-b)^{n}, n \geq 5$, the sum of the $5^{\text {th }}$ and $6^{\text {th }}$ terms is zero. Then $a / b$ equals(a) $(n-5) / 6$
(b) $(n-4) / 5$
(c) $5 /(n-4)$
(d) $6 /(n-5)$

Key Concepts

Binomial Theorem

The Binomial Theorem provides a way to expand an expression raised to a power, such as (a - b)^n, into a sum involving terms of the form C(n, k)a^(n-k)(-b)^k. This theorem is fundamental in understanding how each term’s coefficient is related to combinatorial counts and serves as the starting point for analyzing the properties of any particular term in the expansion.

General Term in Binomial Expansion

The general term in a binomial expansion is given by T_(k+1) = C(n, k)a^(n-k)(-b)^k, where k ranges from 0 to n. This concept is crucial for identifying and extracting specific terms from the expansion by providing a systematic way to express each term based on its position. It allows one to relate the variables and the indices to set up equations as needed.

Equating Specific Term Sums to Zero

When a condition is provided such as the sum of two specific terms being zero, it involves identifying the respective terms using the general term formula and setting up an equation that relates their coefficients and variable powers. This technique involves simplifying the resulting expression by canceling common factors or rearranging terms, leading to a solvable equation for the unknown ratio or variable.

Manipulation of Binomial Coefficients

Understanding the properties of the binomial coefficients, particularly how C(n, k) relates to C(n, k-1), is essential when simplifying expressions generated from the expansion conditions. These relationships often allow for direct cancellation and simplification, which can be used to derive connections between the parameters such as a/b in the given context.

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