document.write( "Question 1016990: an infinite geometric series with second term -8/9 and sum 2. what is the first term? \n" ); document.write( "

Algebra.Com's Answer #633379 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "an infinite geometric series with second term -8/9 and sum 2. what is the first term?
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document.write( "a*r =     (1)    This is your first equation for the second term of GP.\r\n" );
document.write( "                     Here \"a\"is the first term and \"r\" is the common ratio of the GP.\r\n" );
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document.write( " = 2    (2)    This is your second equation for the sum.\r\n" );
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document.write( "From the first equation express r =  and substitute it into the second equation. You will get\r\n" );
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document.write( " = 2,   or\r\n" );
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document.write( " = ,   or\r\n" );
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document.write( " = .\r\n" );
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document.write( "Now multiply both sides by 9a. You will get\r\n" );
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document.write( " = 18a + 16,   or \r\n" );
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document.write( " = .\r\n" );
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document.write( "Apply the quadratic formula to solve this quadratic equation. You will get\r\n" );
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document.write( "two roots  =  and  = -.\r\n" );
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document.write( "The values of \"r\" that correspond to these values of \"a\" in accordance to (1), are\r\n" );
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document.write( " =  :  =   and   =  :  = .\r\n" );
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document.write( "The second  =  has the modulus greater than 1 and therefore generates the \"divergent\" geometric progression. So, the second solution doesn't fit.\r\n" );
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document.write( "Now check the equality (2) for the first solution:  =  :  =  :  =  :  = 2.  OK!\r\n" );
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document.write( "Thus there is a unique GP with the given second term and the given sum.\r\n" );
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document.write( "It is   =  ,   = .\r\n" );
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