document.write( "Question 1191995: a) The first term of an arithmetic progression is 7 and
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\n" ); document.write( "(b) how many terms of the sequence -9,-6,-3 must be taken
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\n" ); document.write( "(c) find the sum of the first 25 even integers.
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Algebra.Com's Answer #823871 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
a) The first term of an arithmetic progression is 7 and
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document.write( "The \"hard\" way is to write them out starting with 7 and \r\n" );
document.write( "adding 2 each time\r\n" );
document.write( "until you have 15 terms:\r\n" );
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document.write( "7,9,11,13,15,17,19,21,23,25,27,29,31,33,35.\r\n" );
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document.write( "The \"easy\" way is to substitute in the formula \"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d\"\r\n" );
document.write( "where a1 = first term = 7 and \r\n" );
document.write( "d = common difference = 2, and n = number of term = 15\r\n" );
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document.write( "\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d\"\r\n" );
document.write( "\"a%5B15%5D=7%2B%2815-1%292\"\r\n" );
document.write( "\"a%5B15%5D=7%2B%2814%292\"\r\n" );
document.write( "\"a%5B15%5D=7%2B28\"\r\n" );
document.write( "\"a%5B15%5D=35\"\r\n" );
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...and the sum of the first 15th term.
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document.write( "The \"hard\" way is to add them up:\r\n" );
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document.write( "7+9+11+13+15+17+19+21+23+25+27+29+31+33+35 - 315.\r\n" );
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document.write( "The \"easy\" way is to substitute in the formula \"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
document.write( "where a1 = first term = 7 and an = a15 = 35.\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
document.write( "\"S%5B15%5D=expr%2815%2F2%29%287%2B35%29\"\r\n" );
document.write( "\"S%5B15%5D=expr%2815%2F2%29%2842%29\"\r\n" );
document.write( "\"S%5B15%5D=315%29\"\r\n" );
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document.write( "There are two formulas for the sum of an arithmetic progression.\r\n" );
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document.write( "If you know the last term, it's easier to use the one we just used,\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%28a%5B1%5D%2Ba%5Bn%5D%29\"\r\n" );
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document.write( "But if you don't know the last term, then use the other one:\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29\"\r\n" );
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document.write( "We don't know the last term, so we use the second formula, with\r\n" );
document.write( "a1=-9 and common difference = \r\n" );
document.write( "(2nd term) - (1st term) =  (-6) - (-9) = -6+9 = 3\r\n" );
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document.write( "\"S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%29\"\r\n" );
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document.write( "We substitute Sn = 66 and solve for n\r\n" );
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document.write( "\"66=expr%28n%2F2%29%282%28-9%29%2B%28n-1%29%283%29%29\"\r\n" );
document.write( "\"66=expr%28n%2F2%29%28-18%2B3n-3%29\"\r\n" );
document.write( "\"66=expr%28n%2F2%29%28-21%2B3n%29\"\r\n" );
document.write( "Multiply both sides by 2\r\n" );
document.write( "\"132=n%28-21%2B3n%29\"\r\n" );
document.write( "\"132=-21n%2B3n%5E2%29\"\r\n" );
document.write( "\"-3n%5E2%2B21n%2B132=0\"\r\n" );
document.write( "Divide through by -3\r\n" );
document.write( "\"n%5E2-7n-44=0\"\r\n" );
document.write( "Factor:\r\n" );
document.write( "\"%28n%2B4%29%28n-11%29=0\"\r\n" );
document.write( "n+4=0;  n-11=0\r\n" );
document.write( "  n=-4;    n=11\r\n" );
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document.write( "n can't be negative, so n = 11. It takes 11 terms to have sum 66.\r\n" );
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document.write( "Edwin
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