document.write( "Question 1170436: the 5th term in a geometric progression is 9 times the 3rd term and the sum of the 6th and 7th terms is 1944. determine the common ratio, the first term the sum of the 4th to 10th terms inclusive
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Algebra.Com's Answer #795338 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "            There are two different progressions and two different answers.\r
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\n" ); document.write( "\n" ); document.write( "The common ratio is   +/- $\"sqrt%289%29\"$ = +/- 3.\r
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\n" ); document.write( "\n" ); document.write( "Case 1.   Common ratio is 3\r
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document.write( "            Then \r\n" );
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document.write( "            a1 is unknown\r\n" );
document.write( "            a2 = 3a1\r\n" );
document.write( "            a3 = 9a1\r\n" );
document.write( "            a5 = 81a1\r\n" );
document.write( "            a6 = 243a1\r\n" );
document.write( "            a7 = 729a1\r\n" );
document.write( "            the last two terms add to 243a1 + 729a1 = 972a1 = 1944\r\n" );
document.write( "            so a1 = 2, and r = 3\r\n" );
document.write( "            -\r\n" );
document.write( "            sum of first 10 terms is   2*(1-3^10)/(-2) = 59048\r\n" );
document.write( "            sum of first  3 terms is   2*(1-3^3)/(-2)  =    26\r\n" );
document.write( "            the sum of the 4th to 10th terms inclusive is  59048  - 26 = 59022\r\n" );
document.write( "            the series is 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Case 2. Common ratio is -3\r
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document.write( "            Then \r\n" );
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document.write( "            a1 is unknown\r\n" );
document.write( "            a2 =   -3a1\r\n" );
document.write( "            a3 =    9a1\r\n" );
document.write( "            a5 =  -81a1\r\n" );
document.write( "            a6 =  243a1\r\n" );
document.write( "            a7 = -729a1\r\n" );
document.write( "            the last two terms add to -729a1 + 243a1 = -486a1 = 1944\r\n" );
document.write( "            so a1 = -4, and r = -3\r\n" );
document.write( "            -\r\n" );
document.write( "            sum of first 10 terms is   (-4)*(1-(-3)^10)/4 = 59048\r\n" );
document.write( "            sum of first  3 terms is   (-4)*(1-(-3)^3)/4  =   -28\r\n" );
document.write( "            the sum of the 4th to 10th terms inclusive is  59048 - (-28) = 59076\r\n" );
document.write( "            the series is -4, 12, -36, 108, -324, 972, -2916, , 8748, -26244.\r\n" );
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