Let the coefficients of third, fourth and fifth terms in the expansion of x+aa2n,x≠0, be in the ratio 12:8:3. Then the term independent of x in the expansion, is equal to

Given binomial expansion is x+ax2nThe ratios of the coefficeints of 3rd, 4th and 5th terms is 12:8:3HenceT3T4=Cn,2a2Cn,3a3=128⇒nn-12nn-1n-26a=32⇒an−2=2T4T5=Cn,3a3Cn,4a4=83⇒an−3=32From the above two equationsn−2n−3=43⇒n=6,a=12To get the independent of x  termEquate the power of x  in general term to zeron−3r=0⇒3r=6⇒r=2Therefore, third term is independent of x