SOLUTION: Find the specified nth term in the expansion of the binomial. (x - 10z)^7, n = 5

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Question 1179417: Find the specified nth term in the expansion of the binomial.
(x - 10z)^7, n = 5
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Find the specified nth term in the expansion of the binomial.
%28x+-+10z%29%5E7, n+=+5=>5th term
%28x+-+10z%29%5E7, n+=+5

%28a%2Bb%29%5En=%28nC%28r-1%29%29+%2Aa%5E%28n-%28r-1%29%29%2Ab%5E%28r-1%29

r=5

%287C%285-1%29%29%28x%29%5E%287-%285-1%29%29%2A%28-10z%29%5E%285-1%29
=%287C4%29%2Ax%5E3%2A%28-10z%29%5E4
=35%2Ax%5E3%2A%2810000z%5E4%29
=350000x%5E3%2Az%5E4



Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the specified nth term in the expansion of the binomial.
(x - 10z)^7, n = 5
~~~~~~~~~~~~~~~ 

They want you determine the 5-th term in the binomial expansion


    %28x-10z%29%5E7 = sum%28C%5B7%5D%5Ek%2Ax%5E%287-k%29%2A%28-10z%29%5Ek%2C+k=0%2C7%29.



     +------------------------------------------------------------------+
     |    The 5-th term  is  at  k=4  (be attentive and notice it:      |
     |    you should take k=4,  NOT  k=5 (!) --- BECAUSE the counter    |
     |           k goes from k = 0)  and is equal to                    |
     +------------------------------------------------------------------+



    C%5B7%5D%5E4%2Ax%5E3%2A%28-10z%29%5E4 = %28%287%2A6%2A5%29%2F%281%2A2%2A3%29%29%2Ax%5E3%2A10%5E4%2Az%5E4 = 350000x%5E3%2Az%5E4.


So the 5-th term is  350000x%5E3%2Az%5E4.      ANSWER

Solved.

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See the lessons
    - Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion
    - Remarkable identities for Binomial Coefficients
    - The Pascal's triangle
    - Solved problems on binomial coefficients
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic
"Binomial expansion, binomial coefficients, Pascal's triangle".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.