document.write( "Question 1117363: if the coefficients of x^k and x^(k+1) in the expansion of (2+3x)^19 are equal, find k \n" ); document.write( "

Algebra.Com's Answer #732431 by ikleyn(52810) $\"\"$ $\"About$

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document.write( "The binomial expansion is this formula\r\n" );
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document.write( " =  +  +  +  + . . . +  + \r\n" );
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document.write( "    (see this Wikipedia article https://en.wikipedia.org/wiki/Binomial_theorem  or my lesson\r\n" );
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document.write( "     Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion  in this site).\r\n" );
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document.write( "For our case, the term containing  is  ,  while the term containing  is  ,\r\n" );
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document.write( "and they want to know at which k the coefficients at    and    are the same:\r\n" );
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document.write( "     = .    (1)\r\n" );
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document.write( "        (notice the difference between \"the terms\" and \"the coefficients\" !)\r\n" );
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document.write( "From (1), you have  \r\n" );
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document.write( "     = ,   or, simplifying the right side,    = .    (2)\r\n" );
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document.write( "Next, use   = ,   = .   \r\n" );
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document.write( "Substitute it into the left side fraction of (2),  and you will get after canceling common factors\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "    2*(k+1) = 3*(19-k)  ====>  2k + 2 = 57 - 3k  ====>  2k + 3k = 57 - 2  ====>  5k = 55  ====>  k =  = 11.\r\n" );
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document.write( "Answer.  k = 11.\r\n" );
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