document.write( "Question 726593: The first 2 terms of a geometric series have a sum of -4. The fourth and fifth terms have a sum of 256. Find the first term and the common ratio of the series. \n" ); document.write( "

Algebra.Com's Answer #444742 by KMST(5328) $\"\"$ $\"About$

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In a geometric series each term equals the previous one times the common ratio:
\n" ); document.write( " $\"b%5Bn%2B1%5D=b%5Bn%5D%2Ar\"$ (recursive formula)
\n" ); document.write( "so that all terms can be calculated as a function of the first term as
\n" ); document.write( " $\"b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29\"$
\n" ); document.write( "
\n" ); document.write( "In this problem,
\n" ); document.write( " $\"b%5B1%5D%2Bb%5B2%5D=-4\"$
\n" ); document.write( " $\"b%5B4%5D%2Bb%5B5%5D=256\"$
\n" ); document.write( "and we want to find $\"b%5B1%5D\"$ and $\"r\"$
\n" ); document.write( "
\n" ); document.write( "MY WAY:
\n" ); document.write( " $\"b%5B4%5D=b%5B1%5D%2Ar%5E3\"$
\n" ); document.write( " $\"b%5B5%5D=b%5B2%5D%2Ar%5E3\"$
\n" ); document.write( " $\"b%5B4%5D%2Bb%5B5%5D=b%5B1%5D%2Ar%5E3%2Bb%5B2%5D%2Ar%5E3\"$
\n" ); document.write( " $\"256=%28b%5B1%5D%2Bb%5B2%5D%29%2Ar%5E3\"$
\n" ); document.write( " $\"256=%28-4%29%2Ar%5E3\"$
\n" ); document.write( " $\"r%5E3=256%2F%28-4%29\"$
\n" ); document.write( " $\"r%5E3=-%282%5E8%2F2%5E2%29\"$
\n" ); document.write( " $\"r%5E3=-2%5E6\"$ --> $\"r=-2%5E2\"$ --> $\"highlight%28r=-4%29\"$
\n" ); document.write( " $\"b%5B1%5D%2Bb%5B1%5D%2Ar=b%5B1%5D%2Bb%5B2%5D\"$
\n" ); document.write( " $\"b%5B1%5D-4b%5B1%5D=-4\"$
\n" ); document.write( " $\"-3b%5B1%5D=-4\"$
\n" ); document.write( " $\"b%5B1%5D=-4%2F%28-3%29\"$ --> $\"highlight%28b%5B1%5D=4%2F3%29\"$
\n" ); document.write( "
\n" ); document.write( "ANOTHER WAY:
\n" ); document.write( " $\"b%5B2%5D=b%5B1%5D%2Ar\"$
\n" ); document.write( " $\"b%5B1%5D%2Bb%5B2%5D=b%5B1%5D%2Bb%5B1%5D%2Ar\"$
\n" ); document.write( " $\"b%5B1%5D%2Bb%5B2%5D=b%5B1%5D%281%2Br%29\"$
\n" ); document.write( " $\"b%5B1%5D%281%2Br%29=-4\"$
\n" ); document.write( " $\"b%5B4%5D=b%5B1%5D%2Ar%5E3\"$
\n" ); document.write( " $\"b%5B5%5D=b%5B1%5D%2Ar%5E4\"$
\n" ); document.write( " $\"b%5B4%5D%2Bb%5B5%5D=b%5B1%5D%2Ar%5E3%2Bb%5B1%5D%2Ar%5E4\"$
\n" ); document.write( " $\"b%5B4%5D%2Bb%5B5%5D=b%5B1%5D%2A%28r%5E3%2Br%5E4%29\"$
\n" ); document.write( " $\"b%5B1%5D%2A%28r%5E3%2Br%5E4%29=256\"$
\n" ); document.write( " $\"system%28b%5B1%5D%281%2Br%29=-4%2Cb%5B1%5D%2A%28r%5E3%2Br%5E4%29=256%29\"$ --> $\"b%5B1%5D%2A%28r%5E3%2Br%5E4%29%2F%28b%5B1%5D%281%2Br%29%29=256%2F%28-4%29\"$ --> $\"%28r%5E3%2Br%5E4%29%2F%281%2Br%29=-64\"$ --> $\"r%5E3%281%2Br%29%2F%281%2Br%29=-64\"$ --> $\"r%5E3=-64\"$ --> $\"r=root%283%2C-64%29\"$ --> $\"highlight%28r=-4%29\"$
\n" ); document.write( " $\"system%28b%5B1%5D%281%2Br%29=-4%2Cr=-4%29\"$ --> $\"b%5B1%5D%281-4%29=-4\"$ --> $\"b%5B1%5D%281-4%29=-4\"$ --> $\"b%5B1%5D=-4%2F%28-3%29\"$ --> $\"highlight%28b%5B1%5D=4%2F3%29\"$ \n" ); document.write( "

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