document.write( "Question 1122510: find the sum of each of the following sequences up to n terms.
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Algebra.Com's Answer #738666 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "find the sum of each of the following sequences up to n terms.
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document.write( "So, we want to find the sum\r\n" );
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document.write( "    R = 8 + 88 + 888 + . . . + 888...8,     (1)\r\n" );
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document.write( "where the last number is written using \"n\" digits of 8.\r\n" );
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document.write( "Consider another sum\r\n" );
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document.write( "    T =  = 9 + 99 + 999 + . . . + 999...9,    (2)\r\n" );
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document.write( "where the last number is written using \"n\" digits of 9.\r\n" );
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document.write( "Add n to T. It is the same as to add \"1\" to each term of the sum (2). Therefore, \r\n" );
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document.write( "    T + n = 10 + 100 + 1000 + . . . + 10^n.      (3)\r\n" );
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document.write( "The sum (3) is the sum of  the first  n  terms of the geometric progression with the first term 10 and the common ratio of 10.  \r\n" );
document.write( "So, the sum (3) is equal to\r\n" );
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document.write( "    T + n =  = .\r\n" );
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document.write( "Thus you have \r\n" );
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document.write( "    T = ,   or\r\n" );
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document.write( "     = ,   which implies\r\n" );
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document.write( "    R = .\r\n" );
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document.write( "It is the final formula and the ANSWER :  R = 8 + 88 + 888 + . . . + 888...8 = .\r\n" );
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document.write( "Check.  Let us take n = 2.  Then from one side,  R = 8 + 88 = 96.\r\n" );
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document.write( "                            From the other side,  R =  =  =  =  =  = 88.   ! Correct !\r\n" );
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document.write( "        Let us take n = 3.  Then from one side,  R = 8 + 88 + 888 = 984.\r\n" );
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document.write( "                            From the other side,  R =  =  =  =  =  = 984.   ! Correct !\r\n" );
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