document.write( "Question 1134773: The sum of the first five terms of an A.P is 36 and the sum of the next four terms is 90. find the A.P \n" ); document.write( "
Algebra.Com's Answer #752249 by ikleyn(52863)\"\" \"About 
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document.write( "Notice that from the condition, the sum of the first NINE terms is 36 + 90 = 126.\r\n" );
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document.write( "Now from the condition we have\r\n" );
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document.write( "      a)  the sum of the first 5 terms is  36,     (1) \r\n" );
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document.write( "      b)  the sum of the first 9 terms is 126.     (2)\r\n" );
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document.write( "For any AP,  the sum of its first n terms is n times the arithmetic mean of the first and the n-th terms - \r\n" );
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document.write( "therefore, for the given AP the arithmetic mean of the first and the 9-th terms is 126/9 =  14.\r\n" );
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document.write( "But in the given case this arithmetic mean of the first and the 9-th terms is nothing else as the 5-th term.\r\n" );
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document.write( "    Thus the 5-th term of the given AP is  14.\r\n" );
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document.write( "Similarly,  for the given AP the arithmetic mean of the first and the 5-th terms is 36/5 =  7.2.\r\n" );
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document.write( "But in this case this arithmetic mean of the first and the 5-th terms is nothing else as the 3-rd term.\r\n" );
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document.write( "    Thus the 3-rd term of the given AP is  7.2.\r\n" );
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document.write( "Between the 3-rd and the 5-th terms there are two equal gaps, each equal to the common difference.\r\n" );
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document.write( "So, the common difference of the given AP is  \"%2814-7.2%29%2F2\" = \"6.8%2F2\" = 3.4.\r\n" );
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document.write( "Now we know that the 3-rd term of the AP is 7.2 and the common difference is 3.4.\r\n" );
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document.write( "Hence, its 1-st term is  7.2 - 2*3.4 = 7.2 - 6.8 = 0.4.\r\n" );
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document.write( "Now we know ALL about this AP:  \"a%5B1%5D\" = 0.4  and  d = 3.4.       ANSWER\r\n" );
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document.write( "CHECK\r\n" );
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document.write( "n       \"a%5Bn%5D\"\r\n" );
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document.write( "1	0.4		\r\n" );
document.write( "2	3.8		\r\n" );
document.write( "3	7.2		\r\n" );
document.write( "4	10.6		\r\n" );
document.write( "5	14	sum from 1 to 5 = 	36       ! Correct !\r\n" );
document.write( "6	17.4		\r\n" );
document.write( "7	20.8		\r\n" );
document.write( "8	24.2		\r\n" );
document.write( "9	27.6	sum from 1 to 9 = 	126      ! Correct !\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice that I solved the problem MENTALLY, without using equations.\r
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\n" ); document.write( "\n" ); document.write( "It was my goal from the very beginning, and therefore I organized my arguments correspondingly.\r
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\n" ); document.write( "\n" ); document.write( "It was how this problem was designed and was intended to be solved.\r
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\n" ); document.write( "\n" ); document.write( "------------------\r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on arithmetic progressions in this site:\r
\n" ); document.write( "\n" ); document.write( "    - Arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Word problems on arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - One characteristic property of arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Math Olimpiad level problem on arithmetic progression \r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Arithmetic progressions\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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