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In the binomial expansion of \left(\sqrt{x}+\frac{1}{2 \sqrt[4]{x}}\right)^nn \in N the coefficients of first, second  and third terms form an A.P. The number of rational terms in the expansion is (Assume that x  is a rational number and \sqrt{x}, \sqrt[4]{x}, are irrational)

 

 

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

\begin{aligned} & \left(\sqrt{x}+\frac{1}{2 \sqrt[4]{x}}\right)^n=\sum_{r=0}^n{ }^n C_r(\sqrt{x})^{n-r} \cdot\left(\frac{1}{2 \sqrt[4]{x}}\right)^r=\sum_{r=0}^n{ }^n C_r \cdot \frac{1}{2^r} x^{\frac{2 n-3 r}{4}} \\ & 1, \frac{n}{2}, \frac{n(n-1)}{8} \text { are in A.P. } \Rightarrow n=8 \end{aligned}

With n = 8, r can take only 3 values i.e. r = 0,4,8

Posted by

Shailly goel

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