SOLUTION: How many terms are there in the expansion of (x-1/x)^18? Is there a term which is independent of x? Find this term if it exists. Also find the middle term in this expansion.

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Question 822118: How many terms are there in the expansion of (x-1/x)^18? Is there a term which is independent of x? Find this term if it exists. Also find the middle term in this expansion.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-1%2Fx%29%5E18%22%22=%22%22%22%22=%22%22%22%22=%22%22sum%28%28matrix%282%2C1%2C18%2Ck%29%29%2A%28x%5Ek%2Ax%5E%28-18%2Bk%29%29%2Ck=0%2C18%29%22%22=%22%22sum%28%28matrix%282%2C1%2C18%2Ck%29%29%2A%28x%5E%282k-18%29%29%2Ck=0%2C18%29

If there is a term independent of x, it will be a term when
the exponent of x is 0, since x0 = 1

So we set the exponent of x equal to zero:

2k-18 = 0
   2k = 18
    k = 9

So the term when k=9 is %28matrix%282%2C1%2C18%2C9%29%29x%5E%282%2A9-18%29%22%22=%22%22%28matrix%282%2C1%2C18%2C9%29%29x%5E0%22%22=%22%22%28matrix%282%2C1%2C18%2C9%29%29%281%29%22%22=%22%22%28matrix%282%2C1%2C18%2C9%29%29 = 48620

Note that %28matrix%282%2C1%2Cn%2Ck%29%29 = C(n,k) = nCk

Since the terms go k=0 to k=18, there are 19 terms and the middle
term will be the 10th term and that is when k=9.  Therefore the term
48620 IS the middle term.

Edwin