document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208641>Question 1208641</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 1,1,3,9 be added respectively to four terms of an AP.,a GP results. Find the four terms of the AP </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847022 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=847022\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>1, 3, 5, 7</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "AP = arithmetic progression<BR>\n" );
document.write( "GP = geometric progression\r<BR>\n" );
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document.write( "We have these 3 sequences<BR>\n" );
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document.write( "<tr><TD >Sequence 1</TD><TD >1</TD><TD >1</TD><TD >3</TD><TD >9</TD></tr><tr><TD >Sequence 2</TD><TD >a</TD><TD >a+d</TD><TD >a+2d</TD><TD >a+3d</TD></tr><tr><TD >Sequence 3</TD><TD >1+a</TD><TD >1+a+d</TD><TD >3+a+2d</TD><TD >9+a+3d</TD></tr>\n" );
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document.write( "Sequence 3 is the sum of sequence 1 and sequence 2.<BR>\n" );
document.write( "Add straight down.\r<BR>\n" );
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document.write( "Because we're told that sequence 3 is geometric, dividing any term over its previous term will get us the common ratio r.\r<BR>\n" );
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document.write( "r = (2nd term)/(1st term) = (1+a+d)/(1+a)<BR>\n" );
document.write( "r = (3rd term)/(2nd term) = (3+a+2d)/(1+a+d) \r<BR>\n" );
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document.write( "Equate those expressions to form this equation<BR>\n" );
document.write( "(1+a+d)/(1+a) = (3+a+2d)/(1+a+d)<BR>\n" );
document.write( "Solving for 'a' leads to a = 0.5d^2 - 1<BR>\n" );
document.write( "I'll let the student do the scratch work.\r<BR>\n" );
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document.write( "Furthermore,<BR>\n" );
document.write( "r = (3rd term)/(2nd term) = (3+a+2d)/(1+a+d) <BR>\n" );
document.write( "r = (4th term)/(3rd term) = (9+a+3d)/(3+a+2d)\r<BR>\n" );
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document.write( "Equating them gives us (3+a+2d)/(1+a+d) = (9+a+3d)/(3+a+2d)\r<BR>\n" );
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document.write( "Plug in a = 0.5d^2 - 1 and we get <BR>\n" );
document.write( "(3+0.5d^2 - 1+2d)/(1+0.5d^2 - 1+d) = (9+0.5d^2 - 1+3d)/(3+0.5d^2 - 1+2d)\r<BR>\n" );
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document.write( "Solving that equation yields d = 2\r<BR>\n" );
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document.write( "Plug d = 2 into a = 0.5d^2 - 1 to get a = 1.\r<BR>\n" );
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document.write( "To summarize:<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "d = 2\r<BR>\n" );
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document.write( "This table<BR>\n" );
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document.write( "<tr><TD >Sequence 1</TD><TD >1</TD><TD >1</TD><TD >3</TD><TD >9</TD></tr><tr><TD >Sequence 2</TD><TD >a</TD><TD >a+d</TD><TD >a+2d</TD><TD >a+3d</TD></tr><tr><TD >Sequence 3</TD><TD >1+a</TD><TD >1+a+d</TD><TD >3+a+2d</TD><TD >9+a+3d</TD></tr>\n" );
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document.write( "Updates to<BR>\n" );
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document.write( "<tr><TD >Sequence 1</TD><TD >1</TD><TD >1</TD><TD >3</TD><TD >9</TD></tr><tr><TD >Sequence 2</TD><TD ><font color=red>1</font></TD><TD ><font color=red>3</font></TD><TD ><font color=red>5</font></TD><TD ><font color=red>7</font></TD></tr><tr><TD >Sequence 3</TD><TD >2</TD><TD >4</TD><TD >8</TD><TD >16</TD></tr>\n" );
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document.write( "Sequence 2 is arithmetic because we add 2 to each term to get the next term.<BR>\n" );
document.write( "The nth term of this sequence is 2n-1 where n is an integer that starts at n = 1.\r<BR>\n" );
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document.write( "Sequence 3 is geometric since dividing any given term over its previous term results in the same common ratio<BR>\n" );
document.write( "4/2 = 2<BR>\n" );
document.write( "8/4 = 2<BR>\n" );
document.write( "16/8 = 2<BR>\n" );
document.write( "Put another way: we double each term to get the next term.<BR>\n" );
document.write( "The nth term of this geometric sequence is 2^n.<BR>\n" );
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