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( "First, informally -- which can be more educational than doing the formal mathematics....<br><BR>\n" );
document.write( "The 38th term is 15 terms after the 23rd term, and the value of the 38th term is 1 more than the value of the 23rd term.  So the common difference in the sequence is 1/15.<br><BR>\n" );
document.write( "The 41st term is 3 terms after the 38th term, so the value of the 41st term is 3 times 1/15 greater than the 38th term.  3*(1/15) = 3/15 = 1/5 = 0.2; the 41st term is 3+0.2 = 3.2.<br><BR>\n" );
document.write( "ANSWER: 3.2<br><BR>\n" );
document.write( "Formally....<br><BR>\n" );
document.write( "if a is the first term and d is the common difference, then...<br><BR>\n" );
document.write( "23rd term is a+22d<BR>\n" );
document.write( "38th term is a+37d<br><BR>\n" );
document.write( "a+37d=3<BR>\n" );
document.write( "a+22d=2<br><BR>\n" );
document.write( "Subtract the second equation from the first<br><BR>\n" );
document.write( "15d=1<BR>\n" );
document.write( "d=1/15<br><BR>\n" );
document.write( "a+22d=2<BR>\n" );
document.write( "a+(22/15)=2=30/15<BR>\n" );
document.write( "a=8/15<br><BR>\n" );
document.write( "The first term is 8/15; the common difference is 1/15.<br><BR>\n" );
document.write( "The 41st term is a+40d = 8/15+40/15 = 48/15 = 16/5 = 3.2<br><BR>\n" );
document.write( "ANSWER (again): 3.2<br><BR>\n" );
document.write( "Note that, if we use the standard formal textbook method for solving the problem, we have to do a lot more work (to find the first term of the sequence) than we did using an informal method.<br><BR>\n" );
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