document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162717>Question 1162717</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine if the statement is true or false. <BR>\n" );
document.write( "If the statement is false, then correct it and make it true.\r<BR>\n" );
document.write( "\n" );
document.write( "The first term in the expansion of (a + b)^999 is a^999. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786556 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=786556\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the statement is correct.<BR>\n" );
document.write( "the first term in the expansion of (a+b)^999 is c(999,0) * a^999 * b^0.<BR>\n" );
document.write( "this becomes (a + b)^999 = 1 * a^999 * 1 which becomes a^999.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the general equation is (a+b)^n = the sum of c(n,k) * a^(n-k) * b^k for k = 0 to k = n\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "note that c(n,k) is equal to n! / (k! * (n-k)!)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "an example will  help clafify the equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "consider (a+b)^1<BR>\n" );
document.write( "in this case, n = 1.<BR>\n" );
document.write( "the formula states that (a+b)^n = the sum of c(n,k) * a^(n-k) * b^k for k = 0 to k = n<BR>\n" );
document.write( "k will go from 0 to 1.<BR>\n" );
document.write( "therefore, the formula becomes:<BR>\n" );
document.write( "(a+b)^1 = c(1,0)*a^1*b^0 + c(1,1)*a^0*b^1<BR>\n" );
document.write( "this becomes 1*a^1 + 1*b^1 which becomes a+b.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this makes sense, since it should be intuitively obvious that (a+b)^1 = (a+b) since any quantity raise to the first power is that quantity itself.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "consider (a+b)^2<BR>\n" );
document.write( "in this case, n = 2.<BR>\n" );
document.write( "the formula states that (a+b)^n = the sum of c(n,k) * a^(n-k) * b^k for k = 0 to k = n<BR>\n" );
document.write( "k will go from 0 to 2.<BR>\n" );
document.write( "therefore, the formula becomes:<BR>\n" );
document.write( "(a+b)^2 = c(2,0)*a^2*b^0 + c(2,1)*a^1*b^1 + c(2,2)*a^0*b^2<BR>\n" );
document.write( "this becomes 1*a^2 + 2*a*b + 1*b^2 which becomes:<BR>\n" );
document.write( "a^2 + 2ab + b^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this makes sense, since (a+b)^2 is equal to (a+b)*(a+b) which is equal to a*(a+b) + b*(a+b) which is equal to a^2 + ab + ab + b^2 which is equal to a^2 + 2ab + b^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can confirm for (a+b)^3 by just following the formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in all cases, however, you can see that the first term is a^n<BR>\n" );
document.write( "when n is 1, the first term is a^1<BR>\n" );
document.write( "then n is 2, the first term is a^2<BR>\n" );
document.write( "when n is 999, the first term is a^999\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "consider (a+b)^999<BR>\n" );
document.write( "the first term is c(999,0)*a^999*b^0 which is equal to a^999.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's some references on the binomial theorem you might find useful.<BR>\n" );
document.write( "pick the reference that makes the most sense to you.<BR>\n" );
document.write( "don't break your brain.<BR>\n" );
document.write( "if you are struggling with the reference, go on to the next one.<BR>\n" );
document.write( "they all should be saying the same thing in different ways, some of which may be more understandable in terms of your thinking.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html\" target = \"_blank\">http://hyperphysics.phy-astr.gsu.edu/hbase/alg3.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://courses.lumenlearning.com/boundless-algebra/chapter/the-binomial-theorem/\" target = \"_blank\">https://courses.lumenlearning.com/boundless-algebra/chapter/the-binomial-theorem/</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://people.richland.edu/james/lecture/m116/sequences/binomial.html\" target = \"_blank\">https://people.richland.edu/james/lecture/m116/sequences/binomial.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/binomial.htm\" target = \"_blank\">https://www.purplemath.com/modules/binomial.htm</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://brilliant.org/wiki/binomial-theorem-n-choose-k/\" target = \"_blank\">https://brilliant.org/wiki/binomial-theorem-n-choose-k/</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"https://mathbitsnotebook.com/Algebra2/Polynomials/POBinomialTheorem.html\" target = \"_blank\">https://mathbitsnotebook.com/Algebra2/Polynomials/POBinomialTheorem.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );