Question 1185059
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find the sixteenth term of (2-1/x)^18.
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            Each such problem  HAS  a  TRAP.          A L W A Y S.  


            In the given problem,  the trap is that the  16-th term  IS  NOT  THE  TERM  with   {{{1/x^16}}}.


            Read my solution below attewntively and consider it as a standard  MANTRA  to solve such problems.


            Following this mantra will allow you to avoid similar traps in the future.



<pre>
The binomial expansion is this formula


    {{{(a+b)^n}}} = {{{a^n}}} + {{{C[n]^1*a^(n-1)*b}}} + {{{C[n]^2*a^(n-2)*b^2}}} + {{{C[n]^3*a^(n-3)*b^3}}} + . . . + {{{C[n]^(n-1)*a^1*b^(n-1)}}} + {{{b^n}}}


In our case,  n = 18,  a = 2,  b = -{{{1/x}}},  therefore, the common term of the binomial expansion in our case is  


    {{{C[18]^k*2^(18-k)*(-1)^k*(1/x)^k}}}, k = 0, 1, 2, 3, 4, 5, . . . , 17, 18.


Notice that k= 0 for the 1-st term;  k= 1 for the 2-nd term, and so on.  Then the 16-th term is at k = 15 


    {{{C[18]^15*2^(18-15)*(-1)^15*(1/x)^15}}} = - {{{((18*17*16)/(1*2*3))*2^3*(1/x)^15}}} = - {{{816*2^3*(1/x)^15}}} = - {{{6528*(1/x^15)}}}.


<U>ANSWER</U>.  The 16-th term is  - {{{6528*(1/x^15)}}}. 
</pre>

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If you want to see many other similar solved problems and to learn the subject wider and deeper, &nbsp;then look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Permutations/Solved-problems-on-binomial-coefficients.lesson>Solved problems on binomial coefficients</A> 

in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this textbook under the topic
"<U>Binomial expansion, binomial coefficients, Pascal's triangle</U>".



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.