document.write( "Question 1195898: https://artofproblemsolving.com/texer/japjtbbh\r
\n" ); document.write( "\n" ); document.write( "Suppose that $\"u_n\"$ is a sequence of real numbers satisfying $\"u_%28n%2B2%29=2u_%28n%2B1%29%2Bu_n\"$ , and that $\"u_3=9\"$ and $\"u_6=128\"$ . What is $\"u_5\"$ ?\r
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Algebra.Com's Answer #828512 by ikleyn(52790) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "Suppose that u_n is a sequence of real numbers satisfying u_(n+1) = 2u_(n+1) + u_n,
\n" ); document.write( "and that u_3=9 and u_6=128. What is u_5 ?
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document.write( "From the basic formula\r\n" );
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document.write( "    u_(n+1) = 2u_(n+1) + u_n\r\n" );
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document.write( "you have, reducing the term u_(n+1) in both sides,\r\n" );
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document.write( "    0 = u_(n+1) + u_n,\r\n" );
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document.write( "or, EQUIVALENTLY,\r\n" );
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document.write( "    u_(n+1) = -u_n.\r\n" );
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document.write( "This formula means that the terms {u_n}, as a sequence, simply change the sign, from any given term to the next one.\r\n" );
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document.write( "Having it in mind, you may conclude that the problem is posed INCORRECTLY, \r\n" );
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document.write( "since u_6 must be -9 then.\r\n" );
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\n" ); document.write( "\n" ); document.write( "ANSWER. The problem is posed INCORRECTLY: it is SELF-CONTRADICTORY.\r
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\n" ); document.write( "\n" ); document.write( "comment from student: Ooops! I revised it here:https://artofproblemsolving.com/texer/japjtbbh\r
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\n" ); document.write( "\n" ); document.write( "My response : In such cases, you should apology and re-submit your problem to the forum, again.\r
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\n" ); document.write( "\n" ); document.write( "I will solve the updated version below.\r
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\n" ); document.write( "\n" ); document.write( "Suppose that u_n is a sequence of real numbers satisfying u_(n+2) = 2u_(n+1)+u_n,
\n" ); document.write( "and that u_3=9 and u_6=128. What is u_5?
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document.write( "Let x be u_4.\r\n" );
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document.write( "My goal is to find x from given data.\r\n" );
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document.write( "Then I will be in position to find u_5.\r\n" );
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document.write( "So, for u_5 I have then, using the basic formula\r\n" );
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document.write( "    u_5 = 2u_4 + u_3 = 2x + 9.\r\n" );
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document.write( "For u_6 I will have\r\n" );
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document.write( "    u_6 = 2*u_5 + u_4 = 2*(2x+9) + x = (4x+18) + x = 5x + 18.\r\n" );
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document.write( "From the other side hand,  u_6 = 128  (given).\r\n" );
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document.write( "It gives me an equation\r\n" );
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document.write( "    5x + 18 = 128\r\n" );
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document.write( "or\r\n" );
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document.write( "    5x      = 128-18 = 110,\r\n" );
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document.write( "     x                 110/5 = 22.\r\n" );
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document.write( "Now  u_5 = 2u_4 + u_3 = 2x + 9 = 2*22 + 9 = 44 + 9 = 53.   \r\n" );
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document.write( "ANSWER.  u_5 = 53.\r\n" );
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