document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209366>Question 1209366</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an arithmetic sequence, the 23rd term is 2, and the 38th term is 3.  What is the 41st term? </SPAN>\n" );
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document.write( "Answer: <font color=red>16/5</font> = <font color=red>3.2</font>\r<BR>\n" );
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document.write( "I'll discuss a slightly different approach to what greenestamps shows in his formal method.<BR>\n" );
document.write( "d = common difference<BR>\n" );
document.write( "23rd term is 2<BR>\n" );
document.write( "24th term is 2+d<BR>\n" );
document.write( "25th term is (2+d)+d = 2+2d<BR>\n" );
document.write( "26th term is (2+2d)+d = 2+3d<BR>\n" );
document.write( "...etc...<BR>\n" );
document.write( "38th term is 2+(38-23)d = 2+15d\r<BR>\n" );
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document.write( "2+15d = 3<BR>\n" );
document.write( "15d = 3-2<BR>\n" );
document.write( "15d = 1<BR>\n" );
document.write( "d = 1/15\r<BR>\n" );
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document.write( "41st term is <BR>\n" );
document.write( "2+(41-23)d = 2+18d = 2+18*(1/15) = <font color=red>16/5</font> = <font color=red>3.2</font>\r<BR>\n" );
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document.write( "--------------------------------------------------------------------------\r<BR>\n" );
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document.write( "Another approach.\r<BR>\n" );
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document.write( "Arithmetic sequences and linear functions are closely connected ideas.\r<BR>\n" );
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document.write( "x = term number = 1,2,3,...<BR>\n" );
document.write( "y = the term itself<BR>\n" );
document.write( "For instance the x = 23rd term is y = 2\r<BR>\n" );
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document.write( "(x,y) = (23,2) and (x,y) = (38,3) are two points on this line.\r<BR>\n" );
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document.write( "Use the slope formula to determine<BR>\n" );
document.write( "m = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "m = (3-2)/(38-23)<BR>\n" );
document.write( "m = 1/15<BR>\n" );
document.write( "The slope is the common difference. This is because it tells us how to get from one term to the next (i.e. one point to the next). This is only when x is a positive whole number.\r<BR>\n" );
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document.write( "Then let's use point-slope form<BR>\n" );
document.write( "y - y1 = m(x - x1)<BR>\n" );
document.write( "y - 2 = (1/15)(x - 23)<BR>\n" );
document.write( "y = (1/15)x - 23/15 + 2<BR>\n" );
document.write( "y = (1/15)x - 23/15 + 30/15<BR>\n" );
document.write( "y = (1/15)x + (-23 + 30)/15<BR>\n" );
document.write( "y = (1/15)x + 7/15<BR>\n" );
document.write( "y = (1/15)*(x+7)<BR>\n" );
document.write( "To verify this equation works, try plugging in x = 23 and you should get y = 2. <BR>\n" );
document.write( "If you plug in x = 38 then you should get y = 3. I'll let the student do these verifications.\r<BR>\n" );
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document.write( "If you plug in x = 41, then it leads to y = <font color=red>16/5</font> = <font color=red>3.2</font><BR>\n" );
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