SOLUTION: if the coefficients of x^k and x^(k+1) in the expansion of (2+3x)^19 are equal, find k

Question 1117363: if the coefficients of x^k and x^(k+1) in the expansion of (2+3x)^19 are equal, find k
Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website!
.

The binomial expansion is this formula


 =  +  +  +  + . . . +  + 


    (see this Wikipedia article https://en.wikipedia.org/wiki/Binomial_theorem  or my lesson

     Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion  in this site).


For our case, the term containing  is  ,  while the term containing  is  ,

and they want to know at which k the coefficients at    and    are the same:

     = .    (1)

        (notice the difference between "the terms" and "the coefficients" !)


From (1), you have  

     = ,   or, simplifying the right side,    = .    (2)


Next, use   = ,   = .   

Substitute it into the left side fraction of (2),  and you will get after canceling common factors

     = ,

    2*(k+1) = 3*(19-k)  ====>  2k + 2 = 57 - 3k  ====>  2k + 3k = 57 - 2  ====>  5k = 55  ====>  k =  = 11.


Answer.  k = 11.

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