Let the coefficients of third, fourth and fifth terms in the expansion of ( x +( a / x 2)) n , x ≠ 0, be in the ratio 12: 8: 3 . Then the term independent of x in the expansion, is equal to .

Question

Let the coefficients of third, fourth and fifth terms in the expansion of ( x +( a / x 2)) n , x ≠ 0, be in the ratio 12: 8: 3 . Then the term independent of x in the expansion, is equal to . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

T r +1= n C r ( x ) n - r (( a / x 2)) r = n C r a T x n -3 r n C 2 a 2: n C 3 a 3: n C 4 a 4=12: 8: 3 After solving n =6, a =(1/2) For term independent of 'x' ⇒ n =3 r r =2 ∴ Coefficient is 6 C 2((1/2))2=(15/4) Nearest integer is 4 .

Q. Let the coefficients of third, fourth and fifth terms in the expansion of (x+x2a​)n,x=0, be in the ratio 12:8:3. Then the term independent of x in the expansion, is equal to ______.

Tr+1​=nCr​(x)n−r(x2a​)r =nCr​aTxn−3r nC2​a2:nC3​a3:nC4​a4=12:8:3 After solving n=6,a=21​ For term independent of 'x' ⇒n=3r r=2 ∴ Coefficient is 6C2​(21​)2=415​ Nearest integer is 4.

Q. Let the coefficients of third, fourth and fifth terms in the expansion of $(x + \frac{a}{x ^{2}})^{n}, x \neq = 0$ , be in the ratio $12 : 8 : 3.$ Then the term independent of $x$ in the expansion, is equal to ______.