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<HTML><HEAD><TITLE>A Simple example-Arithmetic Series Summation</TITLE></HEAD><BODY bgcolor="#FFFFFF"> <a href="../index.html" target="_top"><img src="../icons/usins.gif" alt="Logo" align=right></a><b>Data Structures and Algorithms with Object-Oriented Design Patterns in Python</b><br><A NAME="tex2html1659" HREF="page40.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1657" HREF="page37.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1651" HREF="page38.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A> <A NAME="tex2html1661" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <BR><HR><H2><A NAME="SECTION002120000000000000000">A Simple example-Arithmetic Series Summation</A></H2><P>In this section we apply Axioms <A HREF="page38.html#axiomi"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>, <A HREF="page38.html#axiomii"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A> and <A HREF="page38.html#axiomiii"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>to the analysis of the running time of a program to compute the followingsimple arithmetic series summation<P> <IMG WIDTH=266 HEIGHT=44 ALIGN=BOTTOM ALT="displaymath57669" SRC="img20.gif" ><P>The algorithm to compute this summation is given in Program <A HREF="page39.html#progexamplea"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>.<P><P><A NAME="647"> </A><A NAME="progexamplea"> </A> <IMG WIDTH=575 HEIGHT=142 ALIGN=BOTTOM ALT="program356" SRC="img22.gif" ><BR><STRONG>Program:</STRONG> Program to compute <IMG WIDTH=45 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline57671" SRC="img21.gif" >.<BR><P><P>The executable statements in Program <A HREF="page39.html#progexamplea"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>comprise lines 2-7.Table <A HREF="page39.html#tblsumc"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A> gives the running times of each of these statements.<P><P><A NAME="649"> </A><P> <A NAME="tblsumc"> </A> <DIV ALIGN=CENTER><P ALIGN=CENTER><TABLE COLS=3 BORDER FRAME=HSIDES RULES=GROUPS><COL ALIGN=CENTER><COL ALIGN=CENTER><COL ALIGN=LEFT><TBODY><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> statement </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> time </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP COLSPAN=1> code</TD></TR></TBODY><TBODY><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>2 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=88 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline57639" SRC="img9.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <tt>result = 0</tt></TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 3 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=88 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline57639" SRC="img9.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <tt>i = 1</tt> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 4 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=159 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline57677" SRC="img23.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <tt>i <= n</tt> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 5 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=174 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline57679" SRC="img24.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <tt>result += i</tt> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 6 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=174 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline57679" SRC="img24.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <tt>i += 1</tt> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> 7 </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=96 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline57683" SRC="img25.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> <tt>return result</tt> </TD></TR></TBODY><TBODY><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP>TOTAL </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=227 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline57685" SRC="img26.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD></TR><TR><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> </TD><TD VALIGN=BASELINE ALIGN=CENTER NOWRAP> <IMG WIDTH=230 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline57687" SRC="img27.gif" > </TD><TD VALIGN=BASELINE ALIGN=LEFT NOWRAP> </TD></TR></TBODY><CAPTION ALIGN=BOTTOM><STRONG>Table:</STRONG> Computing the running time of Program <A HREF="page39.html#progexamplea"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>.</CAPTION></TABLE></P></DIV><P><P>Summing the entries in Table <A HREF="page39.html#tblsumc"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A>we get that the running time, <I>T</I>(<I>n</I>),of Program <A HREF="page39.html#progexamplea"><IMG ALIGN=BOTTOM ALT="gif" SRC="../icons/cross_ref_motif.gif"></A> is<P><A NAME="eqnmodelsumc"> </A> <IMG WIDTH=500 HEIGHT=16 ALIGN=BOTTOM ALT="equation387" SRC="img28.gif" ><P>where <IMG WIDTH=238 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline57691" SRC="img29.gif" >and <IMG WIDTH=219 HEIGHT=23 ALIGN=MIDDLE ALT="tex2html_wrap_inline57693" SRC="img30.gif" >.<P><HR><A NAME="tex2html1659" HREF="page40.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="../icons/next_motif.gif"></A> <A NAME="tex2html1657" HREF="page37.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="../icons/up_motif.gif"></A> <A NAME="tex2html1651" HREF="page38.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="../icons/previous_motif.gif"></A> <A NAME="tex2html1661" HREF="page611.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="../icons/index_motif.gif"></A> <P><ADDRESS><img src="../icons/bruno.gif" 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