document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=803180>Question 803180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the general terms of the following sequences<BR>\n" );
document.write( "a. 4,7,12,19,28<BR>\n" );
document.write( "b. 3,-5,7,-9,11<BR>\n" );
document.write( "c. 2,6,12,20,30<BR>\n" );
document.write( "d. 3,-6,12,-24,48 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #484182 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=484182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> find the general terms of the following sequences<BR>\n" );
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document.write( "a.    4, 7, 12, 19, 28\r\n" );
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document.write( "Subtract term by term the sequence of perfect squares:\r\n" );
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document.write( "#1:   1, 4,  9, 16, 25\r\n" );
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document.write( "whose general term is nē\r\n" );
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document.write( "and we get the sequence\r\n" );
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document.write( "#2:   3, 3,  3,  3,  3\r\n" );
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document.write( "whose general term is just 3\r\n" );
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document.write( "So the general term of the given sequence is\r\n" );
document.write( "gotten by reversing the operations we just\r\n" );
document.write( "did with the sequences, with their general\r\n" );
document.write( "terms.  We take the general term of the\r\n" );
document.write( "the last sequence #2: which is 3, and since\r\n" );
document.write( "we subtracted sequence #1 from the given\r\n" );
document.write( "sequence, we will do the opposite and add \r\n" );
document.write( "the general term for #1, which is nē, so\r\n" );
document.write( "the general term of the given sequence (a)\r\n" );
document.write( "is  \r\n" );
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document.write( "a<sub>n</sub> = 3 + nē  \r\n" );
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document.write( "b.    3,-5,7,-9,11\r\n" );
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document.write( "The signs alternate, so\r\n" );
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document.write( "we divide term by term the sequence:\r\n" );
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document.write( "#1:   1,-1,1,-1,1\r\n" );
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document.write( "whose general term is (-1)<sup>n+1</sup>\r\n" );
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document.write( "and get the sequence of positive terms\r\n" );
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document.write( "#2:   3, 5,7, 9, 11\r\n" );
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document.write( "We subtract the sequence\r\n" );
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document.write( "#3:   1, 1,1, 1,  1, whose general term is just 1\r\n" );
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document.write( "and we get the sequence of even integers:\r\n" );
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document.write( "#4:   2, 4,6, 8, 10, whose general term is 2n\r\n" );
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document.write( "So the general term of the given sequence is\r\n" );
document.write( "gotten by reversing the operations we just\r\n" );
document.write( "did with the sequences, with their general\r\n" );
document.write( "terms.  \r\n" );
document.write( "\r\n" );
document.write( "We take the general term of the\r\n" );
document.write( "the last sequence #4: which is 2n, and since\r\n" );
document.write( "we subtracted sequence #3 from sequence #2,\r\n" );
document.write( "we will do the opposite with the general term\r\n" );
document.write( "of #3 which is 1 to 2n, and now we have the\r\n" );
document.write( "general term for #2, which is 2n+1. \r\n" );
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document.write( "Since we divided the given sequence (b) by\r\n" );
document.write( "sequence #1, we do the opposite and multiply\r\n" );
document.write( "2n+1 by the general term of sequence #1 which\r\n" );
document.write( "is  (-1)<sup>n+1</sup>, and get the general\r\n" );
document.write( "term for (b)\r\n" );
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document.write( "a<sub>n</sub> = (-1)<sup>n+1</sup>(2n+1) \r\n" );
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document.write( "c.    2 ,6, 12, 20, 30\r\n" );
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document.write( "Subtract term by term the sequence of perfect squares:\r\n" );
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document.write( "#1:   1, 4,  9, 16, 25\r\n" );
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document.write( "whose general term is nē\r\n" );
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document.write( "and we get the sequence\r\n" );
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document.write( "#2:   1 ,2,  3,  4,  5\r\n" );
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document.write( "whose general term is just n\r\n" );
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document.write( "So the general term of the given sequence is\r\n" );
document.write( "gotten by reversing the operation we just\r\n" );
document.write( "did with the sequences, with their general\r\n" );
document.write( "terms.  We take the general term of the\r\n" );
document.write( "the last sequence #2: which is n, and since\r\n" );
document.write( "we subtracted sequence #1 from the given\r\n" );
document.write( "sequence, we will do the opposite and add \r\n" );
document.write( "the general term for #2, which is nē, so\r\n" );
document.write( "the general term of the given sequence is  \r\n" );
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document.write( "a<sub>n</sub> = n + nē\r\n" );
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document.write( "or if you like, you can factor out n:\r\n" );
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document.write( "a<sub>n</sub> = n(1 + n)  \r\n" );
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document.write( "d.    3,-6,12,-24,48\r\n" );
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document.write( "The signs alternate, so\r\n" );
document.write( "\r\n" );
document.write( "we divide term by term the sequence:\r\n" );
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document.write( "#1:   1,-1, 1, -1, 1\r\n" );
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document.write( "whose general term is (-1)<sup>n+1</sup>\r\n" );
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document.write( "and get the sequence of positive terms\r\n" );
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document.write( "#2:   3, 6,12, 24, 48\r\n" );
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document.write( "We divide each term by the sequence of 3's\r\n" );
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document.write( "#3:   3, 3, 3,  3,  3\r\n" );
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document.write( "whose general term is just 3, and we get\r\n" );
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document.write( "#4:    1, 2, 4, 8, 16 \r\n" );
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document.write( "which is the sequence of powers\r\n" );
document.write( "of 2, starting with the 0 power of 2  \r\n" );
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document.write( "#4:    2<sup>0</sup>,2<sup>1</sup>,2<sup>2</sup>,2<sup>3</sup>, 2<sup>4</sup>\r\n" );
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document.write( "And the sequence of the exponents is\r\n" );
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document.write( "#5:    0, 1, 2, 3,  4\r\n" );
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document.write( "and if we add the sequence of 1's\r\n" );
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document.write( "#6:    1, 1, 1, 1,  1\r\n" );
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document.write( "whose general term is just 1, we get\r\n" );
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document.write( "#7:    1, 2, 3, 4,  5\r\n" );
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document.write( "whose general term is just n.\r\n" );
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document.write( "So the general term of the given sequence is\r\n" );
document.write( "gotten by reversing the operation we just\r\n" );
document.write( "did with the sequences, with their general\r\n" );
document.write( "terms.  \r\n" );
document.write( "\r\n" );
document.write( "We take the general term of the\r\n" );
document.write( "the last sequence #7: which is n, and since\r\n" );
document.write( "we added sequence #6 to sequence #5 to get \r\n" );
document.write( "sequence #7, we will do the opposite and \r\n" );
document.write( "subtract the general term for #6, which is 1, \r\n" );
document.write( "so the general term of sequence #5\r\n" );
document.write( "is n-1.  \r\n" );
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document.write( "#5 was the sequence of exponents in #4, so the\r\n" );
document.write( "general term for sequence #4 is 2<sup>n-1</sup>.\r\n" );
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document.write( "Since we divided sequence #2 by the sequence #3 \r\n" );
document.write( "of 3's to get #4, we will do the opposite and \r\n" );
document.write( "multiply by #3's general term, 3, and so the \r\n" );
document.write( "general term for sequence #2 is 3·2<sup>n-1</sup>.\r\n" );
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document.write( "Since we divided the given sequence (c) by\r\n" );
document.write( "sequence #1 to get sequence #2, we do the opposite\r\n" );
document.write( "and multiply 3·2<sup>n-1</sup> by the general term of\r\n" );
document.write( "sequence #1 which is  (-1)<sup>n+1</sup>, and\r\n" );
document.write( "get the general term for (c):   \r\n" );
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document.write( "a<sub>n</sub> = (-1)<sup>n+1</sup>·3·2<sup>n-1</sup>.\r\n" );
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document.write( "or we might want to put the 3 in front:\r\n" );
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document.write( "a<sub>n</sub> = 3(-1)<sup>n+1</sup>2<sup>n-1</sup>.\r\n" );
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document.write( "Edwin</PRE><BR>\n" );
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