document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1004534>Question 1004534</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of an AP to four terms is 38.<BR>\n" );
document.write( "The sum of the same AP to eight terms is 124.\r<BR>\n" );
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document.write( "How do I work out the first term and the common difference? </SPAN>\n" );
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document.write( "you are given that sum of 4 terms is 38 and sum of 8 terms is 124.\r<BR>\n" );
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document.write( "the formula for sum of an arithmetic sequence is Sn = n/2 * (A1 + An)\r<BR>\n" );
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document.write( "the formula for the nth term of an arithmetic sequence is An = A1 + (n-1)d\r<BR>\n" );
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document.write( "A1 is the first term.<BR>\n" );
document.write( "An is the nth term.<BR>\n" );
document.write( "n is the number of terms.<BR>\n" );
document.write( "d is the common difference.\r<BR>\n" );
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document.write( "since An is equal to A1 + (n-1)d, you can replace An in the formula of Sn = n/2 * (A1 + An with A1 + (n-1)d to get:\r<BR>\n" );
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document.write( "Sn = n/2 * (A1 + An) becomes:\r<BR>\n" );
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document.write( "Sn = n/2 * (A1 + A1 + (n-1)d) which then becomes:\r<BR>\n" );
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document.write( "Sn = n/2 * (2A1 + (n-1)d)\r<BR>\n" );
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document.write( "it is this last formula that will help you find what you are looking for.\r<BR>\n" );
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document.write( "you are given that the sum of the first 4 terms in the sequence is equal to 34.\r<BR>\n" );
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document.write( "the summation formula for that becomes:\r<BR>\n" );
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document.write( "38 = 4/2 * (2A1 + 3d)\r<BR>\n" );
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document.write( "you can simplify this formula to get:\r<BR>\n" );
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document.write( "38 = 4A1 + 6d\r<BR>\n" );
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document.write( "you are also given tha the sum of the first 8 terms in the sequence is equal to 124.\r<BR>\n" );
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document.write( "the summation formula for that becomes:\r<BR>\n" );
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document.write( "124 = 8/2 * (2A1 + 7d)\r<BR>\n" );
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document.write( "You can simplify this to get:\r<BR>\n" );
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document.write( "124 = 8A1 + 28d\r<BR>\n" );
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document.write( "you now have 2 formulas that you need to solve simultaneously.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "38 = 4A1 + 6d<BR>\n" );
document.write( "124 = 8A1 + 28d\r<BR>\n" );
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document.write( "multiply both sides of the first equation by 2 and bring down the second equation as is to get:\r<BR>\n" );
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document.write( "76 = 8A1 + 12d<BR>\n" );
document.write( "124 = 8A1 + 28d\r<BR>\n" );
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document.write( "subtract the first equation from the second equation to get:\r<BR>\n" );
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document.write( "48 = 16d\r<BR>\n" );
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document.write( "solve for d to get:\r<BR>\n" );
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document.write( "d = 3\r<BR>\n" );
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document.write( "now that you know d = 3, you can use either original equation to solve for A1.\r<BR>\n" );
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document.write( "your original equations to use are:\r<BR>\n" );
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document.write( "38 = 4A1 + 6d<BR>\n" );
document.write( "124 = 8A1 + 28d\r<BR>\n" );
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document.write( "you will get A1 = 5.\r<BR>\n" );
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document.write( "your solution is that A1 = 5 and d = 3\r<BR>\n" );
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document.write( "A1 is the first term.<BR>\n" );
document.write( "d is the common difference.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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