document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1110298>Question 1110298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The second, fourth and eight terms of an A.p<BR>\n" );
document.write( "form the first three consecutive terms of a G.P.<BR>\n" );
document.write( "the sum of the third and fifth terms of the A.p is<BR>\n" );
document.write( "equal to 20. Find the (a) first 4 terms of the A.p<BR>\n" );
document.write( "(b) sum of the first 10 terms of the A.p. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #725301 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=725301\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In an A.P. the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5Eth ALIGN=MIDDLE  ALT=\"n%5Eth\"   DISABLED_style= \"max-width:90%; height:auto;\" > term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%2B%28n-1%29d ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%2B%28n-1%29d\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the first term of the A.P. and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the common difference of the A.P.<BR>\n" );
document.write( " <BR>\n" );
document.write( "So, the second, third, fourth, fifth and eight terms are respectively<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=a%2Bd ALIGN=MIDDLE  ALT=\"a%5B2%5D=a%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D=a%2B2d ALIGN=MIDDLE  ALT=\"a%5B3%5D=a%2B2d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D=a%2B3d ALIGN=MIDDLE  ALT=\"a%5B4%5D=a%2B3d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D=a%2B4d ALIGN=MIDDLE  ALT=\"a%5B5%5D=a%2B4d\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B8%5D=a%2B7d ALIGN=MIDDLE  ALT=\"a%5B8%5D=a%2B7d\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The sum of the third and fifth terms is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D%2Ba%5B5%5D=a%2B2d%2Ba%2B4d ALIGN=MIDDLE  ALT=\"a%5B3%5D%2Ba%5B5%5D=a%2B2d%2Ba%2B4d\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D%2Ba%5B5%5D=2a%2B6d ALIGN=MIDDLE  ALT=\"a%5B3%5D%2Ba%5B5%5D=2a%2B6d\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For this problem, that sum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B6d=20 ALIGN=MIDDLE  ALT=\"2a%2B6d=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a%2B3d=10%29 ALIGN=MIDDLE  ALT=\"highlight%28a%2B3d=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one of our equations.<BR>\n" );
document.write( " <BR>\n" );
document.write( "As the ratio of consecutive terms in a G.P. is always the same number,<BR>\n" );
document.write( "if the  second, fourth and eight terms of an A.P.<BR>\n" );
document.write( "form the first three consecutive terms of a G.P. ,<BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D%2Fa%5B2%5D ALIGN=MIDDLE  ALT=\"a%5B4%5D%2Fa%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B8%5D%2Fa%5B4%5D ALIGN=MIDDLE  ALT=\"a%5B8%5D%2Fa%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >=the common ratio of the G.P.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B3d%29%2F%28a%2Bd%29 ALIGN=MIDDLE  ALT=\"%28a%2B3d%29%2F%28a%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B7d%29%2F%28a%2B3d%29 ALIGN=MIDDLE  ALT=\"%28a%2B7d%29%2F%28a%2B3d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2B3d%29%5E2 ALIGN=MIDDLE  ALT=\"%28a%2B3d%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bd%29%28a%2B7d%29 ALIGN=MIDDLE  ALT=\"%28a%2Bd%29%28a%2B7d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2B6ad%2B9d%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2B6ad%2B9d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2B8ad%2B7d%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2B8ad%2B7d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6ad%2B9d%5E2 ALIGN=MIDDLE  ALT=\"6ad%2B9d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8ad%2B7d%5E2 ALIGN=MIDDLE  ALT=\"8ad%2B7d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9d%5E2-7d%5E2=8ad-6ad ALIGN=MIDDLE  ALT=\"9d%5E2-7d%5E2=8ad-6ad\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2d%5E2=2ad ALIGN=MIDDLE  ALT=\"2d%5E2=2ad\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28d=a%29 ALIGN=MIDDLE  ALT=\"highlight%28d=a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now, we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2B3d=10%2Ca=d%29 ALIGN=MIDDLE  ALT=\"system%28a%2B3d=10%2Ca=d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d%2B3d=10%2Ca=d%29 ALIGN=MIDDLE  ALT=\"system%28d%2B3d=10%2Ca=d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%284d=10%2Ca=d%29 ALIGN=MIDDLE  ALT=\"system%284d=10%2Ca=d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28d=2.5%2Ca=d%29 ALIGN=MIDDLE  ALT=\"system%28d=2.5%2Ca=d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28d=2.5%2Ca=2.5%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28d=2.5%2Ca=2.5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a) Knowing the first term, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the common difference, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we can write any terms of the A.P.<BR>\n" );
document.write( "The first 4 terms are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%222.5+%2C+5+%2C+7.5+%2C+10%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%222.5+%2C+5+%2C+7.5+%2C+10%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) You may have been taught a \"formula\" for the sum the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms of an A.P.<BR>\n" );
document.write( "that could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29n%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29n%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or something like that.<BR>\n" );
document.write( "If a teacher likes to see formulas,<BR>\n" );
document.write( "it is a good idea to write the formula the teacher prefers.<BR>\n" );
document.write( "Using that formula, the sum of the first 10 terms calculates as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B10%5D ALIGN=MIDDLE  ALT=\"S%5B10%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2A2.5%2B%2810-1%292.5%2910%2F2 ALIGN=MIDDLE  ALT=\"%282%2A2.5%2B%2810-1%292.5%2910%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2B9%2A2.5%2910%2F2 ALIGN=MIDDLE  ALT=\"%285%2B9%2A2.5%2910%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2B22.5%2910%2F2 ALIGN=MIDDLE  ALT=\"%285%2B22.5%2910%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27.5%2A5 ALIGN=MIDDLE  ALT=\"27.5%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28137.5%29 ALIGN=MIDDLE  ALT=\"highlight%28137.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "I do not memorize formulas, but I remember that<BR>\n" );
document.write( "the average of a number of consecutive terms of an A.P. is the average of the first and last of those terms.<BR>\n" );
document.write( "I can easily list the second , fourth, sixth, eighth, and ten terms as<BR>\n" );
document.write( "5, 10, 15, 20, 25,<BR>\n" );
document.write( "so I can calculate the average of the first 10 terms as <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282.5%2B25%29%2F2 ALIGN=MIDDLE  ALT=\"%282.5%2B25%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the sum of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282.5%2B25%29%2A10%2F2 ALIGN=MIDDLE  ALT=\"%282.5%2B25%29%2A10%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27.5%2A10%2F2 ALIGN=MIDDLE  ALT=\"27.5%2A10%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=275%2F2 ALIGN=MIDDLE  ALT=\"275%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=137.5 ALIGN=MIDDLE  ALT=\"137.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );