document.write( "Question 1162188: In an arithmetic progression, the sum of the first ten terms is 50 and the 5th term is three times the second term. Find the A. First term. B. Common difference. C. The sum of the first 100 terms
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Algebra.Com's Answer #785928 by MathLover1(20850)\"\" \"About 
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\n" ); document.write( "an arithmetic progression:\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D\", \"a%5B1%5D%2Bd\", \"a%5B1%5D%2B2d\",.... where \"a%5B1%5D\" is the first term and \"d\" is Common difference\r
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\n" ); document.write( "\n" ); document.write( "nth term formula:\r
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\n" ); document.write( "\n" ); document.write( "\"a%5Bn%5D=a%5B1%5D%2Bd%28n-1%29\"\r
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\n" ); document.write( "\n" ); document.write( "Denote this partial sum by \"S%5Bn%5D\" . Then\r
\n" ); document.write( "\n" ); document.write( "\"S%5Bn%5D=n%28a%5B1%5D%2Ba%5Bn%5D%29%2F2\" ,where \"n\" is the number of terms, \"a%5B1%5D\" is the first term and \"a%5Bn%5D\" is the last term\r
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\n" ); document.write( "\n" ); document.write( "given: the sum of the first ten terms is \"50+\"\r
\n" ); document.write( "\n" ); document.write( "-> \"n=10\"\r
\n" ); document.write( "\n" ); document.write( "\"50=10%28a%5B1%5D%2Ba%5B10%5D%29%2F2\"\r
\n" ); document.write( "\n" ); document.write( "\"50=5%28a%5B1%5D%2Ba%5B10%5D%29\"\r
\n" ); document.write( "\n" ); document.write( "\"50=5%28a%5B1%5D%2Ba%5B10%5D%29\"\r
\n" ); document.write( "\n" ); document.write( "\"10=a%5B1%5D%E2%80%89%2B%E2%80%89a%5B10%5D\".......solve for \"a%5B1%5D\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=10-a%5B10%5D%E2%80%89\"....\"a%5B10%5D%E2%80%89=a%5B1%5D%2B9d\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=10-%28a%5B1%5D%2B9d%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=10-a%5B1%5D-9d\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D%2Ba%5B1%5D=10-9d\"\r
\n" ); document.write( "\n" ); document.write( "\"2a%5B1%5D=10-9d\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=%2810-9d%29%2F2\".....eq.1\r
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\n" ); document.write( "\n" ); document.write( "the 5th term is three times the second term:\r
\n" ); document.write( "\n" ); document.write( "\"a%5B5%5D=3%2Aa%5B2%5D\"->\"a%5B2%5D=a%5B1%5D%2Bd\" and \"a%5B5%5D=a%5B1%5D%2B4d\"\r
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\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D%2B4d=3%28a%5B1%5D%2Bd%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D%2B4d=3a%5B1%5D%2B3d\"\r
\n" ); document.write( "\n" ); document.write( "\"4d-3d=3a%5B1%5D-a%5B1%5D\"\r
\n" ); document.write( "\n" ); document.write( "\"d=2a%5B1%5D\"......... ..eq.2\r
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\n" ); document.write( "\n" ); document.write( "substitute in eq1\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=%2810-9%282a%5B1%5D%29%29%2F2\".....eq.1\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=5-9%282a%5B1%5D%29%2F2\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=5-9a%5B1%5D\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D%2B9a%5B1%5D=5\"\r
\n" ); document.write( "\n" ); document.write( "\"10a%5B1%5D=5\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D=5%2F10\"\r
\n" ); document.write( "\n" ); document.write( "\"highlight%28a%5B1%5D=1%2F2%29\" -> First term \r
\n" ); document.write( "\n" ); document.write( "go to eq.2\r
\n" ); document.write( "\n" ); document.write( "\"d=2a%5B1%5D\"......... ..eq.2\r
\n" ); document.write( "\n" ); document.write( "\"d=2%281%2F2%29\"\r
\n" ); document.write( "\n" ); document.write( "\"highlight%28d=1%29\"->Common difference\r
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\n" ); document.write( "\n" ); document.write( "nth term formula:\r
\n" ); document.write( "\n" ); document.write( "\"a%5Bn%5D=1%2F2%2B1%28n-1%29\"\r
\n" ); document.write( "\n" ); document.write( "first \"10\" terms are:\r
\n" ); document.write( "\n" ); document.write( "\"1%2F2\", \"1%2F2%2B1\", \"1%2F2%2B2\", \"1%2F2%2B3\",\"1%2F2%2B4\",\"1%2F2%2B5\",\"1%2F2%2B6\", \"1%2F2%2B7\",\"1%2F2%2B8\",\"1%2F2%2B9\"\r
\n" ); document.write( "\n" ); document.write( "or
\n" ); document.write( "\"0.5\", \"1.5\", \"2.5\", \"3.5\",\"4.5\",\"5.5\",\"6.5\", \"7.5\",\"8.5\",\"9.5\"\r
\n" ); document.write( "\n" ); document.write( "check their sum:\r
\n" ); document.write( "\n" ); document.write( "\"S%5B10%5D=10%280.5%2B9.5%29%2F2\"
\n" ); document.write( "\"S%5B10%5D=5%2810%29\"
\n" ); document.write( "\"S%5B10%5D=50\"-> true\r
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\n" ); document.write( "\n" ); document.write( "The sum of the first \"100\" terms:\r
\n" ); document.write( "\n" ); document.write( "first find \"100\"th term: \r
\n" ); document.write( "\n" ); document.write( "\"a%5B1%5D%2B99d\"-> \"0.5%2B99%2A1=99.5\"\r
\n" ); document.write( "\n" ); document.write( "then the sum is:\r
\n" ); document.write( "\n" ); document.write( "\"S%5B100%5D=100%280.5%2B99.5%29%2F2\"\r
\n" ); document.write( "\n" ); document.write( "\"S%5B100%5D=50%28100%29\"\r
\n" ); document.write( "\n" ); document.write( "\"S%5B100%5D=5000\"\r
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