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document.write( "The sum to infinity of a geometric sequence is 27/2 while the sum of the first three terms is 13. Find the sum of the first 5 terms.\r
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document.write( "Thank you for your help.
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document.write( "We have \r\n" );
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document.write( "S = 
= 
+ 
= + 
(1)\r\n" );
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document.write( "Notice that the infinite sum in parentheses of the right-most side is S, again. // It is the KEY IDEA #1.\r\n" );
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document.write( "Based on given info, replace in (1) S by 
and replace the sum of the first three terms by 13. You will get\r\n" );
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document.write( " = 
, or, which is the same\r\n" );
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= 
. (2).\r\n" );
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document.write( "In (2), replace S by as it is given. // It is the KEY IDEA #2.\r\n" );
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document.write( "You will get = 
, which implies 
= 
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document.write( "Hence, q = 
. Thus we just found the common ratio of the progression; it is q = . \r\n" );
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document.write( "Now we are at the finish line.\r\n" );
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document.write( "Similar to (1), we have\r\n" );
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document.write( "S = 
= 
+ 
= 
+ 
(3)\r\n" );
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document.write( "You can re-write it as\r\n" );
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document.write( "S = 
+ 
, (4) // It is the KEY IDEA #3.\r\n" );
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document.write( "where is the sum of the first 5 terms, which is under the question.\r\n" );
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document.write( "Now from (4)\r\n" );
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document.write( " = 
= 
= 
= 
= 
= 
= 
= 
= 
.\r\n" );
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document.write( "Answer. The sum of the first 5 terms of the given GP is .\r
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document.write( " ***** SOLVED *****
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
