document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=838176>Question 838176</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sequence of diagrams is formed by drawing equilateral triangles, each side is 1 cm.<BR>\n" );
document.write( "They give me 3 diagrams.<BR>\n" );
document.write( "1. Triangle <BR>\n" );
document.write( "2. Big triangle with smaller on inside ( makes 4 triangles - 9 lines)<BR>\n" );
document.write( "3. Big with smaller triangles ( 18lines)<BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "The formula for the TOTAL number of one cm lines needed to draw ALL OF THE FIRST n DIAGRAMS is  an^3+bn^2+n<BR>\n" );
document.write( "Find the values of a and b </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #505410 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=505410\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28150%2C150%2C-75%2C75%2C-40%2C110%2C%0D%0Atriangle%28-50%2C0%2C50%2C0%2C0%2C86.6%29%2C%0D%0Alocate%280%2C30%2C1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28250%2C250%2C-125%2C125%2C-125%2C125%2C%0D%0Atriangle%28-50%2C0%2C50%2C0%2C0%2C86.6%29%2Clocate%280%2C30%2C1%29%2C%0D%0Ared%28triangle%28-100%2C-86.6%2C0%2C-86.6%2C-50%2C0%29%29%2C%0D%0Ared%28triangle%28100%2C-86.6%2C0%2C-86.6%2C50%2C0%29%29%2C%0D%0Alocate%28-50%2C-50%2Cred%281%29%29%2Clocate%2850%2C-50%2Cred%282%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28330%2C330%2C-165%2C165%2C-210%2C120%2C%0D%0Atriangle%28-50%2C0%2C50%2C0%2C0%2C86.6%29%2Clocate%280%2C30%2C1%29%2C%0D%0Ared%28triangle%28-100%2C-86.6%2C0%2C-86.6%2C-50%2C0%29%29%2C%0D%0Ared%28triangle%28100%2C-86.6%2C0%2C-86.6%2C50%2C0%29%29%2C%0D%0Alocate%28-50%2C-50%2Cred%281%29%29%2Clocate%2850%2C-50%2Cred%282%29%29%2C%0D%0Agreen%28triangle%28-150%2C-173%2C-50%2C-173%2C-100%2C-86.6%29%29%2Cgreen%28triangle%28-50%2C-173%2C50%2C-173%2C0%2C-86.6%29%29%2C%0D%0Agreen%28triangle%2850%2C-173%2C150%2C-173%2C100%2C-86.6%29%29%2Clocate%28-100%2C-143%2Cgreen%281%29%29%2C%0D%0Alocate%280%2C-143%2Cgreen%282%29%29%2Clocate%28100%2C-143%2Cgreen%283%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "All you have to do to draw each row of triangles is draw the numbered triangles.<BR>\n" );
document.write( "The ones without numbers get formed by the sides of the numbered triangles.<BR>\n" );
document.write( "You draw 1 triangle for row/diagram number 1.<BR>\n" );
document.write( "You add 2 triangles for row 2 in diagram number 2, for a total of 3 triangles.<BR>\n" );
document.write( "You add 3 triangles for row 3 in diagram number 3, for a total of 6 triangles.<BR>\n" );
document.write( "Each triangle drawn takes 3 line segments.<BR>\n" );
document.write( "The first diagram used <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A1=3%29%29%29+segments.%0D%0AThe+second+diagram+used+%7B%7B%7B3%2A3=9 ALIGN=MIDDLE  ALT=\"3%2A1=3%29%29%29+segments.%0D%0AThe+second+diagram+used+%7B%7B%7B3%2A3=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > segments, and the third used <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A6=18 ALIGN=MIDDLE  ALT=\"3%2A6=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The pattern above continues.<BR>\n" );
document.write( "We could calculate formulas for the total number of triangles in the nth diagram, the number of segments in the nth diagram, and the total number of segments in the first diagram, but we do not need to.<BR>\n" );
document.write( "The problem GIVES us the formula (sort of).<BR>\n" );
document.write( "If it is true,<BR>\n" );
document.write( "in the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagram, there are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A1%5E3%2Bb%2A1%5E2%2B1=a%2Bb%2B1 ALIGN=MIDDLE  ALT=\"a%2A1%5E3%2Bb%2A1%5E2%2B1=a%2Bb%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > segments.<BR>\n" );
document.write( "We know that number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2B1=3 ALIGN=MIDDLE  ALT=\"a%2Bb%2B1=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb=3-1 ALIGN=MIDDLE  ALT=\"a%2Bb=3-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb=2 ALIGN=MIDDLE  ALT=\"a%2Bb=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagrams, there should be a total of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2A2%5E3%2Bb%2A2%5E2%2B2=8a%2B4b%2B2 ALIGN=MIDDLE  ALT=\"a%2A2%5E3%2Bb%2A2%5E2%2B2=8a%2B4b%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > segments.<BR>\n" );
document.write( "We know that there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B9=12 ALIGN=MIDDLE  ALT=\"3%2B9=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > segments, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B4b%2B2=12 ALIGN=MIDDLE  ALT=\"8a%2B4b%2B2=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B4b=12-2 ALIGN=MIDDLE  ALT=\"8a%2B4b=12-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8a%2B4b=10 ALIGN=MIDDLE  ALT=\"8a%2B4b=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%2B2b=5 ALIGN=MIDDLE  ALT=\"4a%2B2b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we have a system of two linear equations in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > that is really easy to solve:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%2Bb=2%2C4a%2B2b=5%29 ALIGN=MIDDLE  ALT=\"system%28a%2Bb=2%2C4a%2B2b=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28a=1%2F2%2Cb=3%2F2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28a=1%2F2%2Cb=3%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the total number of segments in the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > diagrams is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E3%2F2%2B3n%5E2%2F2%2Bn ALIGN=MIDDLE  ALT=\"n%5E3%2F2%2B3n%5E2%2F2%2Bn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Just for curiosity, let's se if it works for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E3%2F2%2B3%2A3%5E2%2F2%2B3=27%2F2%2B27%2F2%2B3=27%2B3=30 ALIGN=MIDDLE  ALT=\"3%5E3%2F2%2B3%2A3%5E2%2F2%2B3=27%2F2%2B27%2F2%2B3=27%2B3=30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since we had <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18 ALIGN=MIDDLE  ALT=\"18\"   DISABLED_style= \"max-width:90%; height:auto;\" > segments in the first 3 diagrams, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B9%2B18=30 ALIGN=MIDDLE  ALT=\"3%2B9%2B18=30\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the formula works for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Could we have deduced that formula?<BR>\n" );
document.write( "Yes, laboriously, but what was required was much simpler and easier.<BR>\n" );
document.write( "To arrive at that formula, we could start by calculating the number of segments in the nth diagram<BR>\n" );
document.write( "You add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > triangles for row <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > in diagram number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The total number of triangles drawn to make diagram number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%7D%7D+is%0D%0A%7B%7B%7B1%2B2%2B3%2B%22.....%22%2Bn=n%28n%2B1%29%2F2 ALIGN=MIDDLE  ALT=\"n%7D%7D+is%0D%0A%7B%7B%7B1%2B2%2B3%2B%22.....%22%2Bn=n%28n%2B1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that means<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%28n%2B1%29%2F2=%283%2F2%29n%5E2%2B%283%2F2%29n ALIGN=MIDDLE  ALT=\"3n%28n%2B1%29%2F2=%283%2F2%29n%5E2%2B%283%2F2%29n\"   DISABLED_style= \"max-width:90%; height:auto;\" > 1-cm line segments (sides of those triangles).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%283%2F2%29k%5E2%2B%283%2F2%29k%2C+k=1%2C+k=n+%29 ALIGN=MIDDLE  ALT=\"sum%28%283%2F2%29k%5E2%2B%283%2F2%29k%2C+k=1%2C+k=n+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a sum of polynomials with degree 2,<BR>\n" );
document.write( "so it must be a polynomial of degree 3.<BR>\n" );
document.write( "We write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SUM%5Bn%5D=sum%28%283%2F2%29k%5E2%2B%283%2F2%29k%2C+k=1%2C+k=n+%29=an%5E3%2Bbn%5E2%2Bcn%2Bd ALIGN=MIDDLE  ALT=\"SUM%5Bn%5D=sum%28%283%2F2%29k%5E2%2B%283%2F2%29k%2C+k=1%2C+k=n+%29=an%5E3%2Bbn%5E2%2Bcn%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SUM%5Bn%5D-SUM%5Bn-1%5D=%283%2F2%29n%5E2%2B%283%2F2%29n ALIGN=MIDDLE  ALT=\"SUM%5Bn%5D-SUM%5Bn-1%5D=%283%2F2%29n%5E2%2B%283%2F2%29n\"   DISABLED_style= \"max-width:90%; height:auto;\" > we would find out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );