Arrange the expansion of `(x^(1//2) + (1)/(2x^(1//4)))^n` in decreasing powers of x. Suppose the coefficient of the first three terms form an arithmetic progression. Then the number of terms in the expression having integer powers of x is -
(A) 1
(B) 2
(C) 3
(D) more than 3

Options

A. 1
B. 2
C. 3
D. more than 3

Answer

C

Explanation

`T_(r+1)=.^(n)C_(n)(x^(1//2))^(n-r)(1)/((2x^(1//4))^(r)) = (.^(n)C_(r))/(2^(r))x^((2n-3r)/(4))`
`T_(1), T_(2), T_(3) rarr AP`
` (2.^(n)C_(1))/(2) = .^(n)C_(0) + (.^(n)C_(2))/(2^(2))`
`n-1 = (n(n-1))/(8) rArr n = 8`
`(16-3r)/(4) = "Integers" , r= 0,4,8`