document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205174>Question 1205174</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sequence {𝑎𝑛} is defined by 𝑎1 = 2, 𝑎2 = 3 and 𝑎𝑛+2 = 3𝑎𝑛+1 − 2𝑎𝑛 for all 𝑛 = 1, 2, 3, ….<BR>\n" );
document.write( "Prove by mathematical induction that<BR>\n" );
document.write( "𝑎𝑛 = 2^𝑛−1 + 1 for all 𝑛 = 1, 2, 3, …. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841818 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=841818\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I assume that an = 2^n-1+1 refers to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+2%5E%28n-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+2%5E%28n-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To make \"n-1\" the exponent, surround it in parenthesis.<BR>\n" );
document.write( "You should write an = 2^(n-1)+1\r<BR>\n" );
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document.write( "Base case:<BR>\n" );
document.write( "n = 1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%2B2%5D+=+3%2Aa%5Bn%2B1%5D-2%2Aa%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%2B2%5D+=+3%2Aa%5Bn%2B1%5D-2%2Aa%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%2B2%5D+=+3%2Aa%5B1%2B1%5D-2%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%2B2%5D+=+3%2Aa%5B1%2B1%5D-2%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+3%2Aa%5B2%5D-2%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+3%2Aa%5B2%5D-2%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+3%2A3-2%2A2 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+3%2A3-2%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+5 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+2%5E%28n-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+2%5E%28n-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+2%5E%281-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+2%5E%281-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+2 ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+2%5E%28n-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+2%5E%28n-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D+=+2%5E%282-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5B2%5D+=+2%5E%282-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D+=+3 ALIGN=MIDDLE  ALT=\"a%5B2%5D+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and lastly<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+2%5E%28n-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+2%5E%28n-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+2%5E%283-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+2%5E%283-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+5 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The base case is done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Inductive step:<BR>\n" );
document.write( "Assume that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%5D+=+2%5E%28k-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%5D+=+2%5E%28k-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some integer k such that k > 3.<BR>\n" );
document.write( "The goal is to show <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+2%5E%28k%2B1-1%29%2B1+=+2%5Ek+%2B+1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+2%5E%28k%2B1-1%29%2B1+=+2%5Ek+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > based on that assumption.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A bit of scratch work off to the side<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%5D+=+2%5E%28k-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%5D+=+2%5E%28k-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk-1%5D+=+2%5E%28k-1-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk-1%5D+=+2%5E%28k-1-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk-1%5D+=+2%5E%28k-2%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk-1%5D+=+2%5E%28k-2%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now onto the inductive step<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%2B2%5D+=+3%2Aa%5Bn%2B1%5D-2%2Aa%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%2B2%5D+=+3%2Aa%5Bn%2B1%5D-2%2Aa%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk-1%2B2%5D+=+3%2Aa%5Bk-1%2B1%5D-2%2Aa%5Bk-1%5D ALIGN=MIDDLE  ALT=\"a%5Bk-1%2B2%5D+=+3%2Aa%5Bk-1%2B1%5D-2%2Aa%5Bk-1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reindexing (replace each n with k-1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+3%2Aa%5Bk%5D-2%2Aa%5Bk-1%5D ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+3%2Aa%5Bk%5D-2%2Aa%5Bk-1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+3%2A%282%5E%28k-1%29%2B1%29-2%2A%282%5E%28k-2%29%2B1%29 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+3%2A%282%5E%28k-1%29%2B1%29-2%2A%282%5E%28k-2%29%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitution. Use the inductive assumption shown above.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+3%2A2%5E%28k-1%29%2B3-2%2A2%5E%28k-2%29-2 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+3%2A2%5E%28k-1%29%2B3-2%2A2%5E%28k-2%29-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+3%2A2%5E%28k-1%29-2%2A2%5E%28k-2%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+3%2A2%5E%28k-1%29-2%2A2%5E%28k-2%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+3%2A2%5Ek%2A2%5E%28-1%29-2%2A2%5Ek%2A2%5E%28-2%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+3%2A2%5Ek%2A2%5E%28-1%29-2%2A2%5Ek%2A2%5E%28-2%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+%283%2F2%29%2A2%5Ek-%281%2F2%29%2A2%5Ek%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+%283%2F2%29%2A2%5Ek-%281%2F2%29%2A2%5Ek%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+%283%2F2-1%2F2%29%2A2%5Ek%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+%283%2F2-1%2F2%29%2A2%5Ek%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+%281%29%2A2%5Ek%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+%281%29%2A2%5Ek%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+2%5Ek%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+2%5Ek%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This wraps up the inductive step and the induction proof is complete.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We have shown that the assumption <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%5D+=+2%5E%28k-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bk%5D+=+2%5E%28k-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D+=+2%5Ek+%2B+1 ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D+=+2%5Ek+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the recursive sequence<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5B1%5D+=+2%2C+a%5B2%5D+=+3%2C+a%5Bn%2B2%5D+=+3%2Aa%5Bn%2B1%5D-2%2Aa%5Bn%5D%29 ALIGN=MIDDLE  ALT=\"system%28a%5B1%5D+=+2%2C+a%5B2%5D+=+3%2C+a%5Bn%2B2%5D+=+3%2Aa%5Bn%2B1%5D-2%2Aa%5Bn%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has the closed form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+2%5E%28n-1%29%2B1 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+2%5E%28n-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A spreadsheet can be used to generate a list of examples. <BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );