Question:
In the binomial expansion of $(a - b)^n, n \geq 5,$ a the sum of $5^{th}$ and $6^{th}$ terms is zero, then $\frac{a}{b}$ equals
In the binomial expansion of $(a - b)^n, n \geq 5,$ a the sum of $5^{th}$ and $6^{th}$ terms is zero, then $\frac{a}{b}$ equals
Updated On: Jul 5, 2022
- $\frac{5}{n-4}$
- $\frac{6}{n-5}$
- $\frac{n-5}{6}$
- $\frac{n-4}{5}$
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The Correct Option is D
Solution and Explanation
$^{n}C_{4}\,a^{n-4}\left(-b\right)^{4}+^{n}C_{5}\,a^{n-5}\left(-b\right)^{5}=0$
$\Rightarrow \left(\frac{a}{b}\right)=\frac{n-5+1}{5}.$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.