document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118204>Question 1118204</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 10. Evaluate the following. Show your work using concepts from this unit.\r<BR>\n" );
document.write( "\n" );
document.write( "a) 1 - 3 + 5 - 7 + 9 -11 + ... + 201 - 203\r<BR>\n" );
document.write( "\n" );
document.write( "b) 300 - 299 + 298 - 297 + ... + 100 - 99 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733462 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=733462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I don't know what concepts are in the unit you are studying; the following is the way I would solve these problems.<br><BR>\n" );
document.write( "a) 1 - 3 + 5 - 7 + ... + 201 - 203<br><BR>\n" );
document.write( "To count the number of terms, we can ignore the signs and look at the sequence as an arithmetic sequence with common difference 2.  The number of terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28203-1%29%2F2%2B1+=+102 ALIGN=MIDDLE  ALT=\"%28203-1%29%2F2%2B1+=+102\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Then, taking into account the alternating signs, we can group the 102 terms into 51 pairs, in each of which the sum is -2.  So the sum of this sequence is 51(-2) = -102.<br><BR>\n" );
document.write( "You could also determine the sum less formally by looking at the pattern of the partial sums:<BR>\n" );
document.write( "1-3 = -2<BR>\n" );
document.write( "-2+5 = 3<BR>\n" );
document.write( "3+-7 = -4<BR>\n" );
document.write( "4+9 = 5<BR>\n" );
document.write( "5+-11 = -6<BR>\n" );
document.write( "...<br><BR>\n" );
document.write( "Whenever the last term added is -n, the sum of the sequence is -(n+1)/2.  Since the last term in the sequence is -203, the sum of the sequence is -(203+1)/2 = -102.<br><BR>\n" );
document.write( "b) 300 - 299 + 298 - 297 + ... - 100 + 99<br><BR>\n" );
document.write( "I won't go through the details; the process is exactly the same.<br><BR>\n" );
document.write( "Ignoring signs to count the number of terms, we find there are 202 of them; taking into account the alternating signs, the 202 terms can be seen as 101 pairs, in each of which the sum is 1; then the sum of the sequence is 101*1 = 101.<br><BR>\n" );
document.write( "And again you could get that result by looking at the pattern of the partial sums -- for each two terms in the sequence, the sum of the terms increases by 1, making the sum of the sequence (202/2)*1 = 101*1 = 101. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );