In the expansion of (1+kx)^n, the second and the third term are (-9/2)x and 9x^2 respectively. find the value of k and of n.

Answers:     k = -1/2    n = 9======binomial theorem: the general term of the expansion  is given by    where  is the distance from the first term.  so r=0 for the first term, r=1 for the second term, r=2 for the third term, etc.also,      for  , we have  and  . so the general term for expanding is    we can ignore  because raising 1 to any exponent still gets you 1. multiplying anything by 1 still gets you 1.    ======addressing the second term: the second term is with r = 1. use the general term formula we established for this expansion  with r = 1.    we are given that the second term is  , so      the value of  in terms of n:     so      divide both sides by  to get      ======addressing the third term: the third term is with r = 2. use the general term formula we established for this expansion  with r = 2.    in terms of n is    so    given that  , we set it equal to this general term. divide by x² after    ======we have a system of equations created from our information of the second and third terms:     let us rearrange  so that we can isolate  to substitute into  . this way we deal with the more complicated variable first (two n vs 1 k).    substituting this into equation      to get a common denominator to add  , multiply numerator and denominator of  by  so that denominator becomes      use this value of k in equation  to find      so    k = -1/2    n = 9

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