document.write( "Question 488218: The sum of all terms of an infinite geometric progression is 12, and each term is three times the sum of all terms that follow it. What is the first term of the sequence?\r
\n" ); document.write( "\n" ); document.write( "Please help. I'm not sure if I'm following it correctly but here's what I've got so far:
\n" ); document.write( "Formula: S(infinity)=a1/1-r
\n" ); document.write( "where,
\n" ); document.write( "S(infinity)=12
\n" ); document.write( "a1=3(a2+a3+a4+..an)\r
\n" ); document.write( "\n" ); document.write( "And that's about it. I don't know what to do afterwards. :( \n" ); document.write( "

Algebra.Com's Answer #852360 by ikleyn(52879) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "The sum of all terms of an infinite geometric progression is 12, and each term is three times the sum
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\n" ); document.write( "\n" ); document.write( " Nice problem to wake up your mind.\r
\n" ); document.write( "\n" ); document.write( " I will solve it in a few lines, as simple as I can and as short as it should be.\r
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document.write( "First equation for the infinite sum is\r\n" );
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document.write( "     = 12.     (1)\r\n" );
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document.write( "(I will use standard designations, so I even will not explain them).\r\n" );
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document.write( "The first term is three time the sum of all that follow.\r\n" );
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document.write( "The terms that follow form geometric progressing starting from  with the same \r\n" );
document.write( "common ratio 'r'.  So, we write\r\n" );
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document.write( "     = .     (2)\r\n" );
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document.write( "We can replace here  with .  It will give us\r\n" );
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document.write( "     = .    (3)\r\n" );
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document.write( "Reduce common factor    in both sides.  You will get\r\n" );
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document.write( "    1 = ,\r\n" );
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document.write( "    1 - r = 3r  --->  1 = 3r + r,  --->  1 = 4r  --->  r = 1/4 = 0.25.\r\n" );
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document.write( "Now from equation (1)\r\n" );
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document.write( "     = 12*(1-r) =  =  = 3*3 = 9.\r\n" );
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document.write( "ANSWER.  The first term is   = 9,  the common ratio is  r = 1/4 = 0.25.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved completely in 15 lines, including the answer.\r
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\n" ); document.write( "\n" ); document.write( "Compare it with 58 lines explanations by tutor @Theo.\r
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