document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=175764>Question 175764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all the geometric sequences such that the sum of the first two terms is 2 and the sum of the first three terms is 3.\r<BR>\n" );
document.write( "\n" );
document.write( "I got this:<BR>\n" );
document.write( "a + ar = 2<BR>\n" );
document.write( "a + a(+- 1/&#8730;a) = 2\r<BR>\n" );
document.write( "\n" );
document.write( "I don't know what to do now.  Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #130906 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=130906\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i forget the formulas all the time so i keep having to look them up.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "here's a website that contains geometric series formulas.<BR>\n" );
document.write( "http://www.purplemath.com/modules/series5.htm<BR>\n" );
document.write( "---<BR>\n" );
document.write( "there are others.<BR>\n" );
document.write( "all you have to do is go to yahoo or google and search on geometric series, or geometric sequence, or sum of geometric sequence, etc.  <BR>\n" );
document.write( "---<BR>\n" );
document.write( "after you search a number of times, you'll get the hang of what keywords you should use.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "you can even look in your book.<BR>\n" );
document.write( "since i don't have a book i use the web.<BR>\n" );
document.write( "the web is useful because you get different ways of explaining the material and you can choose the way which is easiest for you to understand.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "my search led me to the following formula.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28+a%5Bi%5D%2C+i=1%2C+n+%29+=+a%5B1%5D%2A%28%281-r%5En%29%2F%281-r%29%29 ALIGN=MIDDLE  ALT=\"sum%28+a%5Bi%5D%2C+i=1%2C+n+%29+=+a%5B1%5D%2A%28%281-r%5En%29%2F%281-r%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---<BR>\n" );
document.write( "my search also led me to the following formula for the nth term of a geometric series.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+a%5B1%5D%2Ar%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---<BR>\n" );
document.write( "you are given that the sum of the first 2 terms is equal to 2.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "a + ar = 2<BR>\n" );
document.write( "---<BR>\n" );
document.write( "you are also given that the sum of the first 3 terms is equal to 3.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "a + ar + ar^2 = 3<BR>\n" );
document.write( "---<BR>\n" );
document.write( "if you subtract the sum of the first 2 terms from the first 3 terms, you get the 3d term.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "a + ar + ar^2 - a - ar = 3 - 2<BR>\n" );
document.write( "which results in:<BR>\n" );
document.write( "ar^2 = 1<BR>\n" );
document.write( "---<BR>\n" );
document.write( "if ar^2 = 1, then<BR>\n" );
document.write( "a = (1/r^2)<BR>\n" );
document.write( "or<BR>\n" );
document.write( "r^2 = (1/a)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "i tried using r^2 = (1/a) but got nowhere.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "i then tried using a = (1/r^2) and got somewhere as follows:<BR>\n" );
document.write( "---<BR>\n" );
document.write( "since the sum of the first 2 terms equals 2, this means that:<BR>\n" );
document.write( "a + ar = 2<BR>\n" );
document.write( "since a = (1/r^2), this also means that:<BR>\n" );
document.write( "(1/r^2) + (1/r^2)*r = 2<BR>\n" );
document.write( "which means that:<BR>\n" );
document.write( "(1/r^2) + 1/r = 2, since r/r^2 = 1/r<BR>\n" );
document.write( "---<BR>\n" );
document.write( "i then multiplied both sides of this equation by r^2 to remove the fractions.<BR>\n" );
document.write( "doing this i got:<BR>\n" );
document.write( "1 + r = 2r^2<BR>\n" );
document.write( "subtracting (1+r) from both sides of the equation, i then got:<BR>\n" );
document.write( "2r^2 - r - 1 = 0<BR>\n" );
document.write( "---<BR>\n" );
document.write( "this factors out to be:<BR>\n" );
document.write( "(2r+1) * (r-1) = 0<BR>\n" );
document.write( "which yields the following results:<BR>\n" );
document.write( "r = 1<BR>\n" );
document.write( "or<BR>\n" );
document.write( "r = -1/2<BR>\n" );
document.write( "---<BR>\n" );
document.write( "either one or both of these values should help solve the equation.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "i tried r = 1 first.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "i already figured out that ar^2 = 1<BR>\n" );
document.write( "if r = 1, this means that:<BR>\n" );
document.write( "a*1^2 = 1<BR>\n" );
document.write( "which means that:<BR>\n" );
document.write( "a = 1.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "if a = 1, and the sum of the first terms is equal to 2, then the 2d term in the sequence must be equal to 1.<BR>\n" );
document.write( "the sum of the first 3 terms is equal to 3, and the 3d term is equal to 1, so this holds up because 1 + 1 + 1 = 3.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "r = 1 is good.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "next i looked at r = -1/2 to see if this is also valid.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "since the 3d terms is equal to 1 and is also equal to ar^2, i solved for:<BR>\n" );
document.write( "ar^2 = 1, using r = (-1/2)<BR>\n" );
document.write( "this became:<BR>\n" );
document.write( "a*(-1/2)^2 = 1<BR>\n" );
document.write( "which became:<BR>\n" );
document.write( "a*(1/4) = 1<BR>\n" );
document.write( "which became:<BR>\n" );
document.write( "a = 4<BR>\n" );
document.write( "---<BR>\n" );
document.write( "if a = 4, and r = -1/2, then the second term would be:<BR>\n" );
document.write( "4*(-1/2) = -2.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "since the first term is 4 and the second term is -2, then the sum of the first 2 terms is 2 which is valid.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "i then took the second term and multiplied it to get the 3d term which should equal 1.<BR>\n" );
document.write( "-2 * (-1/2) = 1 which is equal to the third term.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "since the sum of the first 3 terms is 3, then:<BR>\n" );
document.write( "4 - 2 + 1 should equal 3.<BR>\n" );
document.write( "it does.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "your answer is:<BR>\n" );
document.write( "r can either be 1 or (-1/2).<BR>\n" );
document.write( "if r is 1, then a is 1.<BR>\n" );
document.write( "if r is (-1/2), then a is 4<BR>\n" );
document.write( "---<BR>\n" );
document.write( "what happens if you go out one further?<BR>\n" );
document.write( "take n = 4<BR>\n" );
document.write( "---<BR>\n" );
document.write( "if a = 1 and r = 1 this would be equal to:<BR>\n" );
document.write( "1*1^3 = 1<BR>\n" );
document.write( "---<BR>\n" );
document.write( "if a = 4 and r = (-1/2) this would be equal to:<BR>\n" );
document.write( "1*(-1/2)^3 = -.5<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the fourth terms are not equal, and since the sum of the first 3 terms are equal, it follows that the sum of the first 4 terms will not be equal either.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "in order words, the sum of each of these geometric sequences are only equal when n = 2 or n = 3 and not after, or at least not every time after, and certainly not equal when n = 1.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "since the problem only states that the sum of the first 2 terms and the sum of the first 3 terms are equal, both answers are still valid because they satisfy the given criteria.<BR>\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );