document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1091741>Question 1091741</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help. <BR>\n" );
document.write( "The population of town was 525,000 in 1992 and is growing annually at the rate of 1.75%. Write a recursive sequence Pn for the population. State the first term P1 for the sequence. <BR>\n" );
document.write( "I tried to solve...<BR>\n" );
document.write( "It is geometric sequence, I think. <BR>\n" );
document.write( "So, An=A n-1*r (n & n-1 is smaller letter)<BR>\n" );
document.write( "Pn=P n-1 *(0.0175) This is recursive sequence for Pn.<BR>\n" );
document.write( "Then how do I state the P1?<BR>\n" );
document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #706258 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=706258\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> P1 is equal to 525,000\r<BR>\n" );
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document.write( "r is equal to 1.75% / 100 which is equal to .0175.\r<BR>\n" );
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document.write( "geometric sequence formula is Pn = P1 * r ^ (n-1)\r<BR>\n" );
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document.write( "if you follow that strict interpretation, you would think that Pn = P1 * .0175 ^ (n-1).\r<BR>\n" );
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document.write( "unfortunately that would be wrong.\r<BR>\n" );
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document.write( "r in this case is what you multiply 525,000 by in order to get the next larger term in the sequence.\r<BR>\n" );
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document.write( "that means you have to make r equal to 1 + r, since the next larger number is 525,000 + r * 525,000 which becomes 525,000 * (1 + r).\r<BR>\n" );
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document.write( "your common ratio, in this case, becomes 1 + r which is equal to 1.0175.\r<BR>\n" );
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document.write( "now the formula becomes Pn = 525,000 * 1.0175 ^ (n-1)\r<BR>\n" );
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document.write( "P1 is equal to 525,000.<BR>\n" );
document.write( "common ratio of r is equal to 1.0175<BR>\n" );
document.write( "n is equal to the number of terms in the sequence.\r<BR>\n" );
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document.write( "while this is a correct formula for a geometric sequence, it is not a recursive formula.\r<BR>\n" );
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document.write( "a recursive formula builds on the results of the previous iteration.\r<BR>\n" );
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document.write( "therefore, the recursive formula should be Pn+1 = Pn * 1.0175.\r<BR>\n" );
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document.write( "it can also be stated as Pn = Pn-1 * 1.0175.\r<BR>\n" );
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document.write( "what it's saying is that the current term is equal to the previous term * 1.0175.\r<BR>\n" );
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document.write( "your P1 would be 525,000<BR>\n" );
document.write( "P2 would be P1 * 1.0175<BR>\n" );
document.write( "P3 would be P2 * 1.0175<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "the geometric series formula and the recursive formula are related in the following manner.\r<BR>\n" );
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document.write( "geometric series formula would say that P3 = P1 * 1.075 ^ 2\r<BR>\n" );
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document.write( "recursive formula says that P3 = P2 * 1.0175<BR>\n" );
document.write( "but P2 is equal to P1 * 1.075<BR>\n" );
document.write( "therefore P3 is equal to P1 * 1.075 * 1.075<BR>\n" );
document.write( "this makes P3 equal to P1 * 1.075 ^ 2\r<BR>\n" );
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document.write( "that's exactly what the geometric series formula is saying.\r<BR>\n" );
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document.write( "what's the difference?\r<BR>\n" );
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document.write( "with the geometric series formula, you don't need to know Pn-1 to find Pn.<BR>\n" );
document.write( "you only need to know P1 and the common ratio.\r<BR>\n" );
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document.write( "with the recursive formula, you need to know Pn-1 in order to find Pn.\r<BR>\n" );
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document.write( "hope this helps.<BR>\n" );
document.write( "let me know if you need anything further regarding this problem.\r<BR>\n" );
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document.write( "there can also be confusion between common ratio and interest rate.\r<BR>\n" );
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document.write( "it's true that 1.75% yields an interest rate of .0175.\r<BR>\n" );
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document.write( "that's an interest rate, but it's not a common ratio.\r<BR>\n" );
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document.write( "the common ratio, in this case becomes 1 + the interest rate which makes the common ratio equal to 1.0175.\r<BR>\n" );
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document.write( "different formulas use different definitions of r.\r<BR>\n" );
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document.write( "r can be interpreted as interest rate which gets you .0175.<BR>\n" );
document.write( "r can also be interpreted as common ratio which gets you 1.0175 in this problem.\r<BR>\n" );
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document.write( "same letter but different interpretation.\r<BR>\n" );
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document.write( "your common ratio in this case is 1 plus the interest rate.\r<BR>\n" );
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