In the expansion of \((2^\frac{1}{4}+3^{-\frac{1}{4}})^n\), the ratio of \(5^{th}\) term from start and \(5^{th}\) term form end is \(\sqrt 6 : 1\) , then find \(3^{rd}\) term
- 30\(\sqrt 3\)
- 60\(\sqrt 3\)
- 30
- 50\(\sqrt 3\)
The Correct Option is B
Solution and Explanation
\(\frac{^nC_4 (2^\frac{1}{4})^{n-4}(3^{\frac{-1}{4}})^4}{^nC_4 (2^\frac{-1}{4})^{n-4}(3^{\frac{1}{4}})^4}\) = \(\sqrt 6\)
\((\frac{2^\frac{1}{4}}{3^\frac{-1}{4}})^{(n-8)}\) = \(\sqrt 6\)
\((6)^\frac{n-8}{4}\) = \(\sqrt 6\)
\(n-8=2\)
\(n=10\)
\(T_3 = {^{10}}C_2(2^{\frac{1}{4}})^8 (3^\frac{-1}{4})^2\)
\(= {^{10}}C_2\times(\sqrt2)^4\times\frac{1}{\sqrt3} = 60\sqrt3\)
So, the correct answer is (B): \(60\sqrt 3\)
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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.