If the coefficient of first three terms in the expansion (√ x +(1/2 ⋅ √[4] x )) n where n ∈ N form an arithmetic progression, then find the number of terms in the expansion having integral powers of x.

Question

If the coefficient of first three terms in the expansion (√ x +(1/2 ⋅ √[4] x )) n where n ∈ N form an arithmetic progression, then find the number of terms in the expansion having integral powers of x. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

T r +1= n C r (√ x ) n - r ⋅((1/2 ⋅ √[4] x )) r Also, n C 0, ( n C 1/2), ( n C 2/4) (in that order) are in A.P. ⇒ 2(( n C 1/2))= n C 0+( n C 2/4) ⇒ n =8 ∴ T r +1=( 8 C r /2 r ) ⋅( x )(16-3 r /4) ; r =0,1,2,3,4,5,6,7,8 But ((16-3 r/4)) is an integer for r=0,4,8.

Q. If the coefficient of first three terms in the expansion (x​+2⋅4x​1​)n where n∈N form an arithmetic progression, then find the number of terms in the expansion having integral powers of x.

Tr+1​=nCr​(x​)n−r⋅(2⋅4x​1​)r Also, nC0​,2nC1​​,4nC2​​ (in that order) are in A.P. ⇒2(2nC1​​)=nC0​+4nC2​​⇒n=8 ∴Tr+1​=2r8Cr​​⋅(x)416−3r​;r=0,1,2,3,4,5,6,7,8 But (416−3r​) is an integer for r=0,4,8.

Q. If the coefficient of first three terms in the expansion $(x + \frac{1}{2 \cdot 4 x})^{n}$ where $n \in N$ form an arithmetic progression, then find the number of terms in the expansion having integral powers of $x$ .