(a) for what positive integers $n$ does $\left(x^2+\frac{1}{x}\right)^n$ have a nonzero constant term?

(b) for the values of $n$ that you found in part (a), what is that constant term? (you can leave your answer in the form of a combination.)

a. By the binomial theorem,

which produces a constant term when , or . is even for any choice of , so must be any positive integer for which is an even multiple of 3 i.e. a multiple of 6. Then can be any of the integers in the sequence {6, 12, 18, ...}.

b. Let for some positive integer . Then

and the constant term occurs when , or , or . At this value of , we get the term

in 4 days he will run   12/14

step-by-step explanation:

amount of oatypop that contains the prize = 40%

amount   of oatypop which doesn't contain the prize = (100-40)=60%

probability that hannah will win a prize

    = s +f s +f f s+ff   f s+ff f f s+ infinity.where f= failure,s= success

=   or 60%

probability that hannah only has to buy 3 or less boxes before getting a prize= (s) and ( f s) and (ff s) and (f f f s)

        = = 87.04 %  

step-by-step explanation: