document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197650>Question 1197650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2nd question:<BR>\n" );
document.write( "the terms a,m,n,b form an arithmetic sequence.Express m and n in terms of a and b.(The terms m and n are arithmetic means of the terms a and b.)\r<BR>\n" );
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document.write( "Method I 've tried: d=(b-a)/3 <BR>\n" );
document.write( "Thank you. </SPAN>\n" );
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document.write( "You are on the right track to arrive at d = (b-a)/3<BR>\n" );
document.write( "Nice work so far. \r<BR>\n" );
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document.write( "Given: a,m,n,b form an arithmetic sequence\r<BR>\n" );
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document.write( "What this means is:<BR>\n" );
document.write( "a = first term<BR>\n" );
document.write( "m = second term = a+d<BR>\n" );
document.write( "n = third term = m+d = (a+d)+d = a+2d<BR>\n" );
document.write( "b = fourth term = n+d = (a+2d)+d = a+3d\r<BR>\n" );
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document.write( "Where d is the common difference. It's the length of the gap between terms.<BR>\n" );
document.write( "For a more numeric and concrete example, the arithmetic sequence 1,3,5,7,9 has a common difference of d = 2 as this is the gap between terms (eg: 7-5 = 2)<BR>\n" );
document.write( "Each time we need a new term to the sequence, add on d.\r<BR>\n" );
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document.write( "Let's focus on the line that says<BR>\n" );
document.write( "b = fourth term = n+d = (a+2d)+d = a+3d<BR>\n" );
document.write( "or in short<BR>\n" );
document.write( "b = a+3d\r<BR>\n" );
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document.write( "Let's solve for d.<BR>\n" );
document.write( "b = a+3d<BR>\n" );
document.write( "b-a = 3d<BR>\n" );
document.write( "d = (b-a)/3<BR>\n" );
document.write( "This matches with what you got. You are correct so far.\r<BR>\n" );
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document.write( "Whatever the values of 'a' and b are, we find their difference and split it into 3rds to determine d.<BR>\n" );
document.write( "This is to be expected. Imagine a yard stick where we split into 3 equal pieces. The two cut points will be exactly m and n, which represent 1/3 and 2/3 along the entire stick.<BR>\n" );
document.write( "'a' and b represent the left most and right most parts of the stick before any cuts were made.<BR>\n" );
document.write( "If you're a bit lost, then the visual section below might help clear things up.\r<BR>\n" );
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document.write( "We'll use this value of d to determine m and n in terms of 'a' and b only<BR>\n" );
document.write( "m = a+d<BR>\n" );
document.write( "m = a+(b-a)/3<BR>\n" );
document.write( "m = 3a/3 + (b-a)/3<BR>\n" );
document.write( "m = (3a+(b-a))/3<BR>\n" );
document.write( "<font color=red>m = (2a+b)/3</font><BR>\n" );
document.write( "and<BR>\n" );
document.write( "n = a+2d<BR>\n" );
document.write( "n = a+2(b-a)/3<BR>\n" );
document.write( "n = 3a/3 + (2b-2a)/3<BR>\n" );
document.write( "n = (3a+2b-2a)/3<BR>\n" );
document.write( "<font color=red>n = (a+2b)/3</font><BR>\n" );
document.write( "It's a bit interesting how we have the swap of coefficients when going from 2a+b to a+2b. \r<BR>\n" );
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document.write( "Here's a visual to see why this works out<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/s3XZyEy.png\"><BR>\n" );
document.write( "In the diagram above, (b-a) represents the length on the number line from 'a' to b<BR>\n" );
document.write( "d = (b-a)/3 is the result of splitting this interval into 3 equal pieces.<BR>\n" );
document.write( "The cuts are at m and n\r<BR>\n" );
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document.write( "In the diagram, I made the distance form 'a' to b exactly 9 units (take note of the grid lines on the graph paper). But you can change this total distance to whatever you like and the same concepts will still apply. <BR>\n" );
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