document.write( "Question 675337: find the next term of the sequence {1,4,11,26,57,120,...}? \n" ); document.write( "
Algebra.Com's Answer #419762 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor just gave you the answer. He didn't show how\r\n" );
document.write( "to get it.\r\n" );
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document.write( "1,4,11,26,57,120,...}?\r\n" );
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document.write( "a1 = 1, a2 = 4, a3 = 11, a4 = 26, ...\r\n" );
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document.write( "For a recursion formula, let's try a linear combination \r\n" );
document.write( "of the previous term, an-1, the number of\r\n" );
document.write( "the term, n, and a constant.  We'll let the coefficients be p, q,\r\n" );
document.write( "and the constant be r:\r\n" );
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document.write( "an = p·an-1 + q·n + r\r\n" );
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document.write( "For n = 2\r\n" );
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document.write( "an = p·an-1 + q·n + r\r\n" );
document.write( "a2 = p·a2-1 + q·2 + r\r\n" );
document.write( "a2 = p·a1 + q·2 + r\r\n" );
document.write( "4 = p·1 + 2q + r\r\n" );
document.write( "4 = p + 2q + r\r\n" );
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document.write( "For n = 3\r\n" );
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document.write( "an = p·an-1 + q·n + r\r\n" );
document.write( "a3 = p·a3-1 + q·3 + r\r\n" );
document.write( "a3 = p·a2 + q·3 + r\r\n" );
document.write( "11 = p·4 + 3q + r\r\n" );
document.write( "11 = 4p + 3q + r\r\n" );
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document.write( "For n = 4\r\n" );
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document.write( "an = p·an-1 + q·n + r\r\n" );
document.write( "a4 = p·a4-1 + q·4 + r\r\n" );
document.write( "a4 = p·a3 + q·4 + r\r\n" );
document.write( "26 = p·11 + 4q + r\r\n" );
document.write( "26 = 11p + 4q + r\r\n" );
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document.write( "So we solve this system of equations:\r\n" );
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document.write( " 4 =   p + 2q + r\r\n" );
document.write( "11 =  4p + 3q + r\r\n" );
document.write( "26 = 11p + 4q + r\r\n" );
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document.write( "and get p=2, q=1, r=0\r\n" );
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document.write( "So the recursion becomes:\r\n" );
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document.write( "an = p·an-1 + q·n + r\r\n" );
document.write( "an = 2·an-1 + 1·n + 0\r\n" );
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document.write( "or\r\n" );
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document.write( "an = 2an-1 + n \r\n" );
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document.write( "Now we check to see if this recursion holds\r\n" );
document.write( "for the remaining given terms:\r\n" );
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document.write( "2(11) + 4 =  22 + 4 =  26  yes\r\n" );
document.write( "2(26) + 5 =  52 + 5 =  57  yes\r\n" );
document.write( "2(57) + 6 = 114 + 6 = 120  yes\r\n" );
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document.write( "Indeed it does, so the next term is\r\n" );
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document.write( "      2(120) + 7 = 240 + 7 = 247\r\n" );
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document.write( "Edwin
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