document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203220>Question 1203220</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the sequence for -1, 3 , -5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838597 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52803)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838597\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "The sequence is of three terms, &nbsp;as you see it.\r<BR>\n" );
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document.write( "\"You get what you see\".\r<BR>\n" );
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document.write( "What else we can say ?\r<BR>\n" );
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document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;- it is not arithmetic sequence;\r<BR>\n" );
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document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;- it is not geometric sequence;\r<BR>\n" );
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document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;- it is not exponential;\r<BR>\n" );
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document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;- it is not periodic.\r<BR>\n" );
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document.write( "So, &nbsp;it is not of any sequences, &nbsp;that are studying in school.\r<BR>\n" );
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document.write( "O-o-o ! &nbsp;&nbsp;It is &nbsp;\"alternate\" &nbsp;sequence: &nbsp;it changes the sign from term to term,\r<BR>\n" );
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document.write( "and absolute values form arithmetic sequence; &nbsp;but you can not state &nbsp;(and even can not expect)\r<BR>\n" );
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document.write( "that it will be so with the following terms: &nbsp;&nbsp;you even don't know if the following terms do exist.\r<BR>\n" );
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document.write( "Probably, &nbsp;that is all that we can say.\r<BR>\n" );
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document.write( "Do you see, &nbsp;&nbsp;the question in the post is posed in an \"arbitrary\" form.\r<BR>\n" );
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document.write( "It does not assume a certain answer: &nbsp;the answer also can be in an arbitrary form.\r<BR>\n" );
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document.write( "So, &nbsp;you got my answer in this &nbsp;\"arbitrary\" &nbsp;form, \r<BR>\n" );
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document.write( "which is, &nbsp;from the other point of view, &nbsp;absolutely precise, &nbsp;at the same time.\r<BR>\n" );
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document.write( "Thus, &nbsp;my answer is &nbsp;100% &nbsp;adequate to your question - - - and<BR>\n" );
document.write( "absolutely useless, &nbsp;at the same time.\r<BR>\n" );
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document.write( "So, &nbsp;the question and the answer are of the same worth.\r<BR>\n" );
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document.write( "But I hope, &nbsp;nevertheless, &nbsp;that you have learned something useful for yourself from my sermon.\r<BR>\n" );
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document.write( "This useful is that some questions are nonsensical, &nbsp;and it is better do not ask them.\r<BR>\n" );
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document.write( "After reading this, you may ask me: how then mathematicians study sequences, predict their \"next terms\", <BR>\n" );
document.write( "study their properties and so on ?\r<BR>\n" );
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document.write( "- - - - - - - Sequences come to Math in two ways. - - - - - - - -\r<BR>\n" );
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document.write( "One way is to determine / (to define) the sequence by a rule or by a formula.\r<BR>\n" );
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document.write( "It is the way, &nbsp;how arithmetic and/or geometric progressions came to Math.\r<BR>\n" );
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document.write( "For such sequences, mathematicians develop their properties based on their definitions.\r<BR>\n" );
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document.write( "Another way for a sequence to come to &nbsp;Math is when the sequence describes some properties <BR>\n" );
document.write( "of other mathematical or physical/chemical/biological objects.\r<BR>\n" );
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document.write( "An example of such sequence is the number of diagonals of an n-gon in Geometry.\r<BR>\n" );
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document.write( "In this case the formula for the n-th term is  &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2A%28n-3%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28n%2A%28n-3%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;and in this way, <BR>\n" );
document.write( "we can predict any n-th term.\r<BR>\n" );
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document.write( "One more example is the Fibonacci sequence, &nbsp;which came to &nbsp;Math from consideration <BR>\n" );
document.write( "of population of rabbits &nbsp;(i.e. from &nbsp;Biology).\r<BR>\n" );
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document.write( "For such sequences, mathematicians derive their properties studying the properties of relevant objects in Math or in Science.\r<BR>\n" );
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document.write( "But when somebody writes three numbers and asks \"what sequence is it ?\" - then we only can say <BR>\n" );
document.write( "- - - it is three terms sequence which you see on the paper, or on the screen, or on the desk in classroom.\r<BR>\n" );
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document.write( "Then the question is extremely stupid, and the answer is adequate.\r<BR>\n" );
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document.write( "- - - - - - - - Hope you get my idea. - - - - - - - -\r<BR>\n" );
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document.write( "It is very important to understand it to the end (to the bottom), in order for \r<BR>\n" );
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document.write( "understand, which question about sequences does make sense and which does not.\r<BR>\n" );
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document.write( "In school, nobody teaches students to this issue, so even adult people believe that it is possible <BR>\n" );
document.write( "to predict next term without having solid base for it. It is the same as to believe in our days <BR>\n" );
document.write( "that the Earth is flat.\r<BR>\n" );
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document.write( "The right understanding  only comes with common sense to mature mind (if you're lucky).\r<BR>\n" );
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