document.write( "Question 142221: find the explicit formula
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Algebra.Com's Answer #103540 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
find the explicit formula
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document.write( "The problem is to find a formula that we can use to \r\n" );
document.write( "fill in the empty blanks of this sequence:\r\n" );
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document.write( "___,___,___,___,___,___,-10,___,___,___,___,___,-48\r\n" );
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document.write( "Let's assume it's an arithmetic sequence.  Then we use\r\n" );
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document.write( "an = a1 + (n-1)d\r\n" );
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document.write( "Substitute n=7\r\n" );
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document.write( "an = a1 + (7-1)d\r\n" );
document.write( "a7 = a1 + 6d\r\n" );
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document.write( "when n=7, an = -10, so a7 = -10\r\n" );
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document.write( "Substituting a7 = -10\r\n" );
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document.write( "-10 = a1 + 6d\r\n" );
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document.write( "Also,\r\n" );
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document.write( "an = a1 + (n-1)d\r\n" );
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document.write( "Substitute n=13\r\n" );
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document.write( "a13 = a1 + (13-1)d\r\n" );
document.write( "a13 = a1 + 12d\r\n" );
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document.write( "when n=13, an = -48, so a13 = -48\r\n" );
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document.write( "Substituting a13 = -48\r\n" );
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document.write( "-48 = a1 + 12d\r\n" );
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document.write( "So we have the system of two equations and\r\n" );
document.write( "two unknowns\r\n" );
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document.write( "-10 = a1 + 6d\r\n" );
document.write( "-48 = a1 + 12d\r\n" );
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document.write( "So we solve that system and get:\r\n" );
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document.write( "a1 = 28, d = -19/3\r\n" );
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document.write( "So to get the formula, we take\r\n" );
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document.write( "an = a1 + (n-1)d\r\n" );
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document.write( "And substitute those values of a1 and d\r\n" );
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document.write( "an = \"28+%2B+%28n-1%29%28-19%2F3%29\"\r\n" );
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document.write( "an = \"28+-%2819%2F3%29n+%2B+19%2F3\"\r\n" );
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document.write( "an = \"84%2F3+-+%2819n%29%2F3+%2B+19%2F3\"\r\n" );
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document.write( "an = \"%28103-19n%29%2F3\"\r\n" );
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document.write( "That's the formula.  We can now fill in the blanks:\r\n" );
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document.write( "28,65/3,46/3,9,8/3,-11/3,-10,-49/3,-68/3,-29,-106/3,-48.\r\n" );
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document.write( "That's the solution your teacher wants, most likely.\r\n" );
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document.write( "However, it is also possible that it could be a \r\n" );
document.write( "geometric sequence as well:\r\n" );
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document.write( "Let's assume it's a geometric sequence.  Then we use\r\n" );
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document.write( "an = a1rn-1\r\n" );
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document.write( "Substitute n=7\r\n" );
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document.write( "a7 = a1r7-1\r\n" );
document.write( "a7 = a1r6\r\n" );
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document.write( "when n=7, an = -10, so a7 = -10\r\n" );
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document.write( "Substituting a7 = -10\r\n" );
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document.write( "-10 = a1r6\r\n" );
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document.write( "Also,\r\n" );
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document.write( "an = a1rn-1\r\n" );
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document.write( "Substitute n=13\r\n" );
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document.write( "a13 = a1r13-1\r\n" );
document.write( "a13 = a1r12\r\n" );
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document.write( "when n=13, an = -48, so a13 = -48\r\n" );
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document.write( "Substituting a13 = -48\r\n" );
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document.write( "-48 = a1r12\r\n" );
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document.write( "So we have the system of two equations and\r\n" );
document.write( "two unknowns\r\n" );
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document.write( "-10 = a1r6\r\n" );
document.write( "-48 = a1r12\r\n" );
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document.write( "So we solve that system and get:\r\n" );
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document.write( "a1 = \"-25%2F12\", d = \"root%286%2C%2848%29%29\"\r\n" );
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document.write( "So to get the formula, we take\r\n" );
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document.write( "an = a1rn-1\r\n" );
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document.write( "And substitute those values of a1 and d\r\n" );
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document.write( "an = \"%28-25%2F12%29%28root%286%2C48%29%29%5E%28n-1%29\"\r\n" );
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document.write( "That's the formula.  We can now fill in the blanks,\r\n" );
document.write( "but I'll just give their decimal values:\r\n" );
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document.write( "-2.083,-2.706,-3.514,-4.564,-5.928,-7.699,\r\n" );
document.write( "-10,-12.99,-16.87,-21.91,-28.46,-36.96,-48.\r\n" );
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document.write( "Actually, arithmetic and geometric sequences are only 2 kinds of\r\n" );
document.write( "many sequences.  So technically there are infinitely many answers to\r\n" );
document.write( "this problem, but only one arithmetic sequence solution, and only one\r\n" );
document.write( "geometric sequence solution. \r\n" );
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document.write( "Edwin
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