document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199475>Question 1199475</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The 7th term is 56 and the 12th term is -1792 of the geometric <BR>\n" );
document.write( "sequence. Find the ratio and the first term. Assume the <BR>\n" );
document.write( "ratios are equal. </SPAN>\n" );
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document.write( "r = common ratio\r<BR>\n" );
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document.write( "x = 7th term<BR>\n" );
document.write( "xr = 8th term<BR>\n" );
document.write( "xr^2 = 9th term<BR>\n" );
document.write( "xr^3 = 10th term<BR>\n" );
document.write( "xr^4 = 11th term<BR>\n" );
document.write( "xr^5 = 12th term\r<BR>\n" );
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document.write( "Each time we need a new term, we multiply the previous term by the common ratio r.<BR>\n" );
document.write( "The gap from the 7th term and the 12th term is 12-7 = 5, which is the exponent for xr^5.\r<BR>\n" );
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document.write( "In this case, x = 56 is the seventh term.<BR>\n" );
document.write( "This means xr^5 updates to 56r^5<BR>\n" );
document.write( "Set this equal to the given 12th term -1792\r<BR>\n" );
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document.write( "56r^5 = -1792<BR>\n" );
document.write( "r^5 = -1792/56<BR>\n" );
document.write( "r^5 = -32<BR>\n" );
document.write( "From here you can use mental math to determine r = -2 because (-2)^5 = -32\r<BR>\n" );
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document.write( "Or you could say this<BR>\n" );
document.write( "r^5 = -32<BR>\n" );
document.write( "r = (-32)^(1/5)<BR>\n" );
document.write( "r = -2<BR>\n" );
document.write( "The exponent of 1/5 represents the 5th root.\r<BR>\n" );
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document.write( "The nth term of a geometric sequence is a*r^(n-1)<BR>\n" );
document.write( "a = first term<BR>\n" );
document.write( "r = common ratio<BR>\n" );
document.write( "n = term number\r<BR>\n" );
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document.write( "Let's plug r = -2 and n = 7.<BR>\n" );
document.write( "We'll use the fact the 7th term is 56.\r<BR>\n" );
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document.write( "nth term = a*r^(n-1)<BR>\n" );
document.write( "7th term = a*r^(7-1)<BR>\n" );
document.write( "56 = a*(-2)^(7-1)<BR>\n" );
document.write( "56 = a*64<BR>\n" );
document.write( "a = 56/64<BR>\n" );
document.write( "a = 7/8<BR>\n" );
document.write( "In decimal form this would be exactly 0.875, but I'll stick to the fraction form.\r<BR>\n" );
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document.write( "The nth term is therefore,<BR>\n" );
document.write( "a(n) = (7/8)*(-2)^(n-1)\r<BR>\n" );
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document.write( "To verify this answer, plug n = 7 and you should get a(7) = 56.<BR>\n" );
document.write( "Also, you should find that a(12) = -1792.<BR>\n" );
document.write( "I'll let you do these verification steps.\r<BR>\n" );
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document.write( "Answers: \r<BR>\n" );
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document.write( "First term = <font color=red>7/8</font><BR>\n" );
document.write( "Common ratio = <font color=red>-2</font><BR>\n" );
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