document.write( "Question 546647: 1.If the third term of of a Geometric Progression is the square of the first and the fifth term is 64,find the series. \n" ); document.write( "
Algebra.Com's Answer #356072 by Edwin McCravy(20056)\"\" \"About 
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1.If the third term of a Geometric Progression is the square of the first and the fifth term is 64,find the series.
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document.write( "This is what we are given, where a1 represents the first term.\r\n" );
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document.write( "a1, ___, a1², ___, 64, ...\r\n" );
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document.write( "Let's assume that r represents the common ratio. and so the second term\r\n" );
document.write( "is the first term times r:\r\n" );
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document.write( "a1, a1r, a1², ___, 64, ...\r\n" );
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document.write( "Since the third term is the second term times r, we have the equation\r\n" );
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document.write( "(a1r)(r) = (a1)²\r\n" );
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document.write( "a1r² = (a1)²\r\n" );
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document.write( "We divide both sides by a1 \r\n" );
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document.write( "    r² = a1\r\n" );
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document.write( "The square of the common ratio and the first term are the same, so now the\r\n" );
document.write( "geometric progression starts with r² and each successive term\r\n" );
document.write( "is the precding one multiplied by r, so we have:\r\n" );
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document.write( "r2, r3, r4, r5, r6 = 64\r\n" );
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document.write( "So we solve for r:\r\n" );
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document.write( "r6 = 64\r\n" );
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document.write( "Use the principle of even (sixth) roots, \r\n" );
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document.write( "r = ±2\r\n" );
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document.write( "So there are two solutions one with r = 2 and one with r = -2\r\n" );
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document.write( "Using r = 2\r\n" );
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document.write( "So since \r\n" );
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document.write( "r² = a1\r\n" );
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document.write( "The first term a1 = 2² = 4\r\n" );
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document.write( "The nth term is \r\n" );
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document.write( "an = a1rn-1 = 4(2)n-1) = 2²(2)n-1) = 22+n-1 = 2n+1\r\n" );
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document.write( "and the progression is\r\n" );
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document.write( "4, 8, 16, 32, 64, ...\r\n" );
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document.write( "Using r = -2\r\n" );
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document.write( "So since \r\n" );
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document.write( "r² = a1\r\n" );
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document.write( "The first term a1 = (-2)² = 4\r\n" );
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document.write( "The nth term is \r\n" );
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document.write( "an = a1(-2)n-1 = 4(-2)n-1) = 4(-1·2)n-1) = 4(-1)n-12n-1 =  2²(-1)n-12n-1 =  (-1)n-12n-1+2 =  (-1)n-12n+1\r\n" );
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document.write( "and the progression is\r\n" );
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document.write( "4, -8, 16, -32, 64, -+ ...\r\n" );
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document.write( "So the nth term is either given by\r\n" );
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document.write( "an = 2n+1\r\n" );
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document.write( "or by:\r\n" );
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document.write( "an = (-1)n-12n+1\r\n" );
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document.write( "Edwin
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