document.write( "Question 828588: What is the sequence to 2 7 22 67 202 \n" ); document.write( "
Algebra.Com's Answer #499431 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "2 7 22 67 202\r\n" );
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document.write( "Each term is 1 more than 3 times the preceding term.\r\n" );
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document.write( "A recursive rule would be \"to get the next term, multiply the previous term\r\n" );
document.write( "by 3 then add 1\": \r\n" );
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document.write( "\"a%5B1%5D=2\", \"a%5Bn%2B1%5D=3a%5Bn%5D%2B1\"\r\n" );
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document.write( "But let's see if we can get the general term:\r\n" );
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document.write( "We make the sequences of differences between successive \r\n" );
document.write( "terms to see if they follow a pattern\r\n" );
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document.write( "  2    7-2 =   5 = 5×30\r\n" );
document.write( "  7   22-7 =  15 = 5×31\r\n" );
document.write( " 22  67-22 =  45 = 5×32\r\n" );
document.write( " 67 202-67 = 135 = 5×33\r\n" );
document.write( "202\r\n" );
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document.write( "1st term = 2\r\n" );
document.write( "2nd term = 2+5×30 = 7\r\n" );
document.write( "3rd term = 2+5×30+5×31 = 22\r\n" );
document.write( "4th term = 2+5×30+5×31+5×32 = 67\r\n" );
document.write( "5th term = 2+5×30+5×31+5×32+5×33 = 202\r\n" );
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document.write( "We see a pattern and we would suppose that the next term is\r\n" );
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document.write( "6th term = 2+5×30+5×31+5×32+5×33+5×34 = 607\r\n" );
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document.write( "So we assume that the general term is:\r\n" );
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document.write( "kth term = 2+5×30+5×31+5×32+···+5×3k-2\r\n" );
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document.write( "or factoring 5 out of all those with factor 5:\r\n" );
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document.write( "kth term = 2+5(30+31+32+···+3k-2)\r\n" );
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document.write( "The terms in parentheses is the sum of a geometric sequence with \r\n" );
document.write( "\"a%5B1%5D\"=1, \"a%5Bn%5D\"=3k-2,  r=3,   \r\n" );
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document.write( "Sn = \"%28a%5B1%5D-r%2Aa%5Bn%5D%29%2F%281-r%29\" \r\n" );
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document.write( "\"%281-3%2A3%5E%28k-2%29%29%2F%281-3%29\" = \"%281-3%5E%28k-1%29%29%2F%28-2%29\" = \"%28-%28-1%2B3%5E%28k-1%29%29%29%2F%28-2%29\" = \"%28-1%2B3%5E%28k-1%29%29%2F2\" = \"%283%5E%28k-1%29-1%29%2F2\"\r\n" );
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document.write( "Now let's go back the the general term \r\n" );
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document.write( "kth term = 2+5(30+31+32+···+3k-2)\r\n" );
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document.write( "and substitute for the terms in the parentheses:\r\n" );
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document.write( "kth term = 2+5\"%28%283%5E%28k-1%29-1%29%2F2%29\" = \"2%2F1%2Bexpr%285%2F1%29%28%283%5E%28k-1%29-1%29%2F2%29\" = \"4%2F2%2B%285%283%5E%28k-1%29-1%29%29%2F2%29\" = \r\n" );
document.write( "\"4%2F2%2B%285%2A3%5E%28k-1%29-5%29%2F2%29\" = \"%284%2B5%2A3%5E%28k-1%29-5%29%2F2%29\" = \"%285%2A3%5E%28k-1%29-1%29%2F2%29\"\r\n" );
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document.write( "So if you want to call it the nth term instead of the kth term, just\r\n" );
document.write( "use n instead of k:\r\n" );
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document.write( "\"a%5Bn%5D\"\"%22%22=%22%22\"\"%285%2A3%5E%28n-1%29-1%29%2F2%29\"\r\n" );
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document.write( "You can crank out as many terms as you like.  Here are the first 20:\r\n" );
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document.write( "2, 7, 22, 67, 202, 607, 1822, 5467, 16402, 49207, 147622, 442867, 1328602,\r\n" );
document.write( "3985807, 11957422, 35872267, 107616802, 322850407, 968551222, 2905653667.\r\n" );
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document.write( "Edwin
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