document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=765337>Question 765337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The first term of geometric sequence is a^-4, the second term is a^x. If the 8th term is a^52, what is the value of x?<BR>\n" );
document.write( "Please advise me how to solve this,<BR>\n" );
document.write( "I figured out a^x= ar and ar^7 = 52, and I don't know where it goes from here.<BR>\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #466087 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=466087\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a.1 = the first term in the sequence.<BR>\n" );
document.write( "a.2 = second term in the sequence.<BR>\n" );
document.write( "a.n = nth term in the sequence.<BR>\n" );
document.write( "you are given that a.1 = a^-4 (first term in the sequence).<BR>\n" );
document.write( "you are also given that a.2 = a^x (second term in the sequence).<BR>\n" );
document.write( "you are also given that a.8 = a^52 (eighth term in the sequence).\r<BR>\n" );
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document.write( "first thing you want to do is solve for r.<BR>\n" );
document.write( "you can use the first and second terms in the sequence to do this.<BR>\n" );
document.write( "first term is a^-4.<BR>\n" );
document.write( "second term is a^x<BR>\n" );
document.write( "formula for second term is:<BR>\n" );
document.write( "a.2 = a.1 * r^(n-1)<BR>\n" );
document.write( "in the second term of the sequence, n is equal to 2.<BR>\n" );
document.write( "in the first term of the sequence, n is equal to 1.<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( "a.2 = a.1 * r^(2-1) which becomes a.2 = a.1 * r^1 which becomes a.2 = a.1 * r<BR>\n" );
document.write( "your formula becomes:<BR>\n" );
document.write( "a.2 = a.1 * r (second term in the sequence is equal to a.1 * r).<BR>\n" );
document.write( "since a.2 is equal to a^x and a.1 is equal to a^-4, this formula becomes:<BR>\n" );
document.write( "a^x = a^-4 * r<BR>\n" );
document.write( "you can now solve for r in terms of a as follows:<BR>\n" );
document.write( "divide both sides of this equation by a^-4 to get:<BR>\n" );
document.write( "r = a^x / a^-4<BR>\n" );
document.write( "the laws of exponent arithmetic make this equal to:<BR>\n" );
document.write( "r = a^(x-(-4)) which becomes:<BR>\n" );
document.write( "r = a^(x+4)<BR>\n" );
document.write( "you now have the value of r.<BR>\n" );
document.write( "r is equal to a^(x+4)<BR>\n" );
document.write( "you are given that the 8th term in the sequence is equal to a^52.<BR>\n" );
document.write( "the 8th term in the sequence is given by the formula:<BR>\n" );
document.write( "a.8 = a.1 * r^(8-1) which becomes:<BR>\n" );
document.write( "a.8 = a.1 * r^7)<BR>\n" );
document.write( "you know that a.1 is equal to a^-4 and you know that r = a^(x+4) and you know that a.8 is equal to a^52, so you can susbtitute in the formula for a.8 to get:<BR>\n" );
document.write( "a^52 = a^-4 * (a^(x+4))^7<BR>\n" );
document.write( "by the rules of exponent arithmetic, a^(x+4))^7 becomes a^(7*(x+4)) which becomes<BR>\n" );
document.write( "a^(7x+28).<BR>\n" );
document.write( "your formula becomes:<BR>\n" );
document.write( "a^52 = a^-4 * a^(7x+28).<BR>\n" );
document.write( "by the rules of exponent arithmetic, a^-4 * a^(7x+28) is equal to a^(-4 + 7x + 28) which is equal to a^(7x + 24).<BR>\n" );
document.write( "your formula becomes:<BR>\n" );
document.write( "a^52 = a^(7x + 24)<BR>\n" );
document.write( "this can only be true if 52 = 7x + 24.<BR>\n" );
document.write( "now you can solve for x as follows:<BR>\n" );
document.write( "subtract 28 from both sides of that equation to get:<BR>\n" );
document.write( "52 - 24 = 7x<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "28 = 7x<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 4<BR>\n" );
document.write( "that's your answer.<BR>\n" );
document.write( "x = 4<BR>\n" );
document.write( "you can confirm your answer is good by substituting for x in all the equations that use it.<BR>\n" );
document.write( "before you do that, however, solve for r with x = 4.<BR>\n" );
document.write( "when x = 4, r = a^(x+4) becomes a^8.<BR>\n" );
document.write( "the first term in your sequence is equal to a^-4 (given).<BR>\n" );
document.write( "the second term in your sequence is equal to a^x which is equal to a^4 because x = 4.<BR>\n" );
document.write( "since a.2 = a.1 * r, this means that a.2 = a.1 * a^8 because r = a^8.<BR>\n" );
document.write( "since a.2 = a^4 and a.1 = a^-4, then this formula becomes:<BR>\n" );
document.write( "a^4 = a^-4 * a^8.<BR>\n" );
document.write( "by the rules of exponent arithmetic, a^-4 * a^8 = a^(-4 + 8) = a^4 and you get:<BR>\n" );
document.write( "a^4 = a^4.<BR>\n" );
document.write( "by similar manipulations you can solve for a.8 as well.<BR>\n" );
document.write( "a.8 = a.1 * r^(8-1) which is equal to a.1 * r^7.<BR>\n" );
document.write( "since r = a^8, this formula becomes:<BR>\n" );
document.write( "a.8 = a.1 * (a^8)^7 which becomes:<BR>\n" );
document.write( "a.8 = a.1 * a^56<BR>\n" );
document.write( "since a.1 = a^-4 and a^8 = a^52, this formula becomes:<BR>\n" );
document.write( "a^52 = a^-4 * a^56<BR>\n" );
document.write( "by the rules of exponent arithmetic, a^-4 * a^56 = a^(-4 + 56) which becomes a^-52.<BR>\n" );
document.write( "your formula becomes:<BR>\n" );
document.write( "a^52 = a^52 which is true, confirming the value of x = 4 is good.,\r<BR>\n" );
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