document.write( "Question 981006: If S1,S2,S3 be the sum of n,2n,3n terms respectively of an A.P.Then (a)S3=S1+S2 \r
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\n" ); document.write( "\n" ); document.write( "(b)S3=2(S1+S2) (c)S3=3(S2-S1) (d)S3=3(S1+S2)\r
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\n" ); document.write( "\n" ); document.write( "If Sn denotes the sum of first n terms of an A.P and S2n=3Sn,then S3n/Sn=\r
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Algebra.Com's Answer #602149 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
If \"S%5B1%5D\" , \"S%5B2%5D\" , and \"S%5B3%5D\" are the sums of \"n\" , \"2n\" , \"3n\" terms respectively of an A.P., then \"highlight%28S%5B3%5D=3%28S%5B2%5D-S%5B1%5D%29%29\" .\r
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\n" ); document.write( "\n" ); document.write( "ONE WAY TO SOLVE THE PROBLEM is to applying somewhat involved algebraic manipulations,
\n" ); document.write( "and (possibly memorized) formulas derived from (possibly forgotten, possibly never fully understood) concepts:
\n" ); document.write( "For an A.P. with first term \"a%5B1%5D\" and common difference \"d\" ,
\n" ); document.write( "Sum of first \"m\" terms =\"2a%5B1%5D%2B%28m-1%29d%29m%2F2\" <---> \"S%5Bm%5D=%282ma%5B1%5D%2Bm%28m-1%29d%29%2F2\"
\n" ); document.write( "For \"m=n\" , we have \"S%5B1%5D=%282na%5B1%5D%2Bn%28n-1%29d%29%2F2=%282na%5B1%5D%2B%28n%5E2-n%29d%29%2F2\" .
\n" ); document.write( "For \"m=2n\" , we have \"S%5B2%5D=%284na%5B1%5D%2B2n%282n-1%29d%29%2F2=%284na%5B1%5D%2B%284n%5E2-2n%29d%29%2F2\" .
\n" ); document.write( "For \"m=3n\" , we have \"S%5B3%5D=%286na%5B1%5D%2B3n%283n-1%29d%29%2F2=%286na%5B1%5D%2B%289n%5E2-3n%29d%29%2F2\" .
\n" ); document.write( " ,
\n" ); document.write( "so the three choices with \"S%5B1%5D%2BS%5B2%5D\" do not work.
\n" ); document.write( "On the other hand,
\n" ); document.write( " ,
\n" ); document.write( "so \r
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\n" ); document.write( "\n" ); document.write( "ANOTHER WAY TO LOOK AT THE PROBLEM
\n" ); document.write( "involves visualizing what those sums are made of and how they are related,
\n" ); document.write( "relying more on concepts than on formulas,
\n" ); document.write( "and using easier arithmetic rather than advanced algebra.
\n" ); document.write( "Unfortunately, describing a clear reasoning based on basic concepts often takes a lot of words (or a lot of algebraic expressions).
\n" ); document.write( "Let the terms of the A.P. be \"a%5B1%5D\" , \"a%5B2%5D=a%5B1%5D%2Bd\" , \"a%5B3%5D=a%5B1%5D%2B2d\" , etc.
\n" ); document.write( "\"S%5B1%5D=a%5B1%5D%2Ba%5B2%5D%2B%22...%22%2Ba%5Bn%5D\"
\n" ); document.write( "\"S%5B2%5D=S%5B1%5D%2Ba%5Bn%2B1%5D%2Ba%5Bn%2B2%5D%2B%22...%22%2Ba%5B2n%5D\"
\n" ); document.write( "\"S%5B3%5D=S%5B2%5D%2Ba%5B2n%2B1%5D%2Ba%5B2n%2B2%5D%2B%22...%22%2Ba%5B3n%5D\"
\n" ); document.write( "We know that each term is the one before plus \"d\" ,
\n" ); document.write( "and the difference between two terms is \"d\" times the difference between the term numbers,
\n" ); document.write( "as in \"a%5Bk%5D=a%5B1%5D%2B%28k-1%29d\" <---> \"a%5Bk%5D-a%5B1%5D%2B%28k-1%29d\" ,
\n" ); document.write( "so \"a%5Bn%2B1%5D=a%5B1%5D%2Bnd\" ,
\n" ); document.write( "but that applies more generally, not just when one of those two terms is \"a%5B1%5D\" ,
\n" ); document.write( "so \"a%5Bn%2B2%5D=a%5B2%5D%2Bnd\" , \"a%5Bn%2B3%5D=a%5B3%5D%2Bnd\" , and so on up to
\n" ); document.write( "\"a%5B2n%5D=a%5Bn%2Bn%5D=a%5Bn%5D%2Bnd\" .
\n" ); document.write( "So, .
\n" ); document.write( "We can write \"S%5B2%5D\" as a function of \"X=S%5B1%5D\" and \"Y=n%5E2d\" :
\n" ); document.write( "\"S%5B2%5D=2S%5B1%5D%2Bn%5E2d\" <--> \"S%5B2%5D=2X%2BY\" .
\n" ); document.write( "By a similar reasoning,
\n" ); document.write( "since \"a%5B2n%2B1%5D=a%5B1%5D%2B2nd\" , \"a%5B2n%2B2%5D=a%5B2%5D%2B2nd\" , and so on, up to \"a%5B3n%5D=a%5B2n%2Bn%5D=a%5Bn%5D%2B2nd\" ,
\n" ); document.write( " ,
\n" ); document.write( "so \"S%5B3%5D=S%5B2%5D%2BS%5B1%5D%2B2n%5E2d\" ,
\n" ); document.write( "and \"system%28S%5B2%5D=2S%5B1%5D%2Bn%5E2d%2CS%5B3%5D=S%5B2%5D%2BS%5B1%5D%2B2n%5E2d%29\" ---> \"S%5B3%5D=2S%5B1%5D%2Bn%5E2d%2BS%5B1%5D%2B2n%5E2d=3S%5B1%5D%2B3n%5E2d\" .
\n" ); document.write( "We can write \"S%5B3%5D\" as a function of \"X=S%5B1%5D\" and \"Y=n%5E2d\" :
\n" ); document.write( "\"S%5B3%5D=3S%5B1%5D%2B3n%5E2d\" <--> \"S%5B3%5D=3X%2B3Y\" .
\n" ); document.write( "Looking at \"system%28S%5B1%5D=X%2CS%5B2%5D=2X%2BY%2CS%5B3%5D=3X%2B3Y%29\" ,
\n" ); document.write( "it is clear that none of the formulas with \"S%5B1%5D%2BS%5B2%5D=3X%2BY\" will work,
\n" ); document.write( "but \"S%5B2%5D-S%5B1%5D=X%2BY\"--->\"3%28S%5B2%5D-S%5B1%5D%29=3%28X%2BY%29=3X%2B3Y=S%5B3%5D\"\r
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\n" ); document.write( "\n" ); document.write( "NOW FOR THE EASY PROBLEM:
\n" ); document.write( "If \"Sn\" denotes the sum of first \"n\" terms of an A.P.,
\n" ); document.write( "and \"S%5B2n%5D=3S%5Bn%5D\" , then \"S%5B3n%5D%2FS%5Bn%5D\"= ?
\n" ); document.write( "We proved above that if \"S%5B1%5D\" , \"S%5B2%5D\" , and \"S%5B3%5D\" are the sums of \"n\" , \"2n\" , \"3n\" terms respectively of an A.P., then \"highlight%28S%5B3%5D=3%28S%5B2%5D-S%5B1%5D%29%29\" .
\n" ); document.write( "Using the easier symbols from the problem above,
\n" ); document.write( "the new question is
\n" ); document.write( "if \"S%5B2%5D=3S%5B1%5D\" , then \"S%5B3%5D%2FS%5B1%5D\"= ?
\n" ); document.write( "\"S%5B3%5D=3%28S%5B2%5D-S%5B1%5D%29=3S%5B2%5D-3S%5B1%5D\" , so
\n" ); document.write( "\"system%28S%5B3%5D=3S%5B2%5D-3S%5B1%5D%2CS%5B2%5D=3S%5B1%5D%29\"--->\"S%5B3%5D=3%283S%5B1%5D%29-3S%5B1%5D=9S%5B1%5D-3S%5B1%5D=6S%5B1%5D\"--->\"S%5B3%5D%2FS%5B1%5D=6S%5B1%5D%2FS%5B1%5D=highlight%286%29\" \n" ); document.write( "
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