1. Tardigrade
  2. Question
  3. Mathematics
  4. Arrange the expansion of (x(1/2)+(1/2 x(1)4))n in decreasing powers of x. Suppose the coefficient of the first three terms (taken in that order) form an arithmetic progression. Then the number of terms in the expansion having integral powers of x, is

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Q. Arrange the expansion of $\left(x^{\frac{1}{2}}+\frac{1}{2 x^{\frac{1}{4}}}\right)^n$ in decreasing powers of $x$. Suppose the coefficient of the first three terms (taken in that order) form an arithmetic progression. Then the number of terms in the expansion having integral powers of $x$, is

Binomial Theorem

Solution:

$T _{ r +1}={ }^{ n } C _{ r }\left( x ^{\frac{1}{2}}\right)^{ n - r }\left(\frac{1}{2 x ^{\frac{1}{4}}}\right)^{ r }=\frac{{ }^{ n } C _{ r }}{2^{ r }} x ^{\frac{2 n -3 r }{4}}$
As, $T_1, T_2, T_3$ (in A.P.)
$\Rightarrow 2\left(\frac{{ }^{ n } C _1}{2}\right)={ }^{ n } C _0+\frac{{ }^{ n } C _2}{2^2} \Rightarrow n =8 $
$\therefore \frac{16-3 r }{4}=\text { integer, when } r =0,4,8$