document.write( "Question 1192532: Several points are marked on a line. Tina then marked another point between each two adjacent points on the line. He repeated this process a further 3 times. There are now 225 points marked on the line. How many points were marked on the line initially? \n" ); document.write( "

Algebra.Com's Answer #824407 by ikleyn(52835) $\"\"$ $\"About$

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\n" ); document.write( "Several points are marked on a line.
\n" ); document.write( "Tina then marked another point between each two adjacent points on the line.
\n" ); document.write( "He repeated this process a further 3 times.
\n" ); document.write( "There are now 225 points marked on the line.
\n" ); document.write( "How many points were marked on the line initially?
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document.write( "Every time, as we start with n points on the line, we insert (n-1) points in between, and we get \r\n" );
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document.write( "    n + (n-1) = 2n-1 points after inserting.\r\n" );
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document.write( "So, first inserting leads us from n points to 2n-1 points.\r\n" );
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document.write( "Repeating such inserting one more time leads us from 2n-1 points to 2*(2n-1)-1 = 4n-3 points.\r\n" );
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document.write( "Repeating such inserting second time leads us from 2n-1 points to 2*(4n-3)-1 = 8n-7 points.\r\n" );
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document.write( "Repeating such inserting third time leads us from 8n-7 points to 2*(8n-7)-1 = 16n-15 points.\r\n" );
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document.write( "From the given part of the problem, we have this equation\r\n" );
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document.write( "    16n - 15 = 225.\r\n" );
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document.write( "It gives\r\n" );
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document.write( "    n =  =  = 15.\r\n" );
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document.write( "ANSWER.  Originally, there were 15 points.\r\n" );
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document.write( "CHECK.  15 points give 2*15-1 = 29 pints after first inserting.\r\n" );
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document.write( "        It gives 2*29-1 =  57 points after first repeating.\r\n" );
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document.write( "        It gives 2*57-1 = 113 points after second repeating.\r\n" );
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document.write( "        It gives 2*113-1 = 225 points after third repeating.    ! Correct !\r\n" );
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