document.write( "Question 1040859: Q: An increase in geometric progression has positive terms.The sum of the first eight terms is seventeen times the sum of the first four terms.The seven term is 960.The ninth term is? \n" ); document.write( "

Algebra.Com's Answer #655878 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
In geometric progression, each term is equal to the one before times a certain number called the common ratio, $\"r\"$ .
\n" ); document.write( "If we call the first term $\"b%5B1%5D\"$ ,
\n" ); document.write( "everything else can be calculated from $\"b%5B1%5D\"$ and $\"r\"$ .
\n" ); document.write( "The $\"n%5Eth\"$ term is $\"b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29\"$ ,
\n" ); document.write( "and the sum of the first $\"n\"$ terms is
\n" ); document.write( " $\"S%5Bn%5D=b%5B1%5D%2A%28%28r%5En-1%29%2F%28r-1%29%29\"$ .
\n" ); document.write( "If we knew $\"b%5B1%5D\"$ and $\"r\"$ ,
\n" ); document.write( "we could calculate
\n" ); document.write( "the sum of the first $\"4\"$ terms as
\n" ); document.write( " $\"S%5B4%5D=b%5B1%5D%2A%28%28r%5E4-1%29%2F%28r-1%29%29\"$ ;
\n" ); document.write( "the sum of the first $\"8\"$ terms as
\n" ); document.write( " $\"S%5B8%5D=b%5B1%5D%2A%28%28r%5E8-1%29%2F%28r-1%29%29\"$ ;
\n" ); document.write( "the $\"9%5Eth\"$ term as
\n" ); document.write( " $\"b%5B7%5D=b%5B1%5D%2Ar%5E%287-1%29=b%5B1%5D%2Ar%5E6\"$ , and
\n" ); document.write( "the $\"9%5Eth\"$ term as
\n" ); document.write( " $\"b%5B9%5D=b%5B1%5D%2Ar%5E%289-1%29=b%5B1%5D%2Ar%5E8\"$ .
\n" ); document.write( "The problem tells us that
\n" ); document.write( "\"the sum of the first eight terms is seventeen times the sum of the first four terms\" ,
\n" ); document.write( "so $\"S%5B8%5D=17%2AS%5B4%5D\"$ or $\"b%5B1%5D%2A%28%28r%5E8-1%29%2F%28r-1%29%29=17%2Ab%5B1%5D%2A%28%28r%5E4-1%29%2F%28r-1%29%29\"$ , and
\n" ); document.write( "\"the seventh term is $\"960\"$ \" ,
\n" ); document.write( "so $\"b%5B7%5D=960\"$ or $\"b%5B1%5D%2Ar%5E6=960\"$ .
\n" ); document.write( "From the two equations above, we can find $\"b%5B1%5D\"$ and $\"r\"$ ,
\n" ); document.write( "and from $\"b%5B1%5D\"$ and $\"r\"$ ,
\n" ); document.write( "we can find $\"b%5B9%5D=b%5B1%5D%2Ar%5E8\"$ .
\n" ); document.write( "Alternately, $\"b%5B9%5D=b%5B8%5D%2Ar=b%5B7%5D%2Ar%5E2\"$ .
\n" ); document.write( "
\n" ); document.write( " $\"b%5B1%5D%2A%28%28r%5E8-1%29%2F%28r-1%29%29=17%2Ab%5B1%5D%2A%28%28r%5E4-1%29%2F%28r-1%29%29\"$
\n" ); document.write( " $\"%28r%5E8-1%29=17%2A%28r%5E4-1%29\"$
\n" ); document.write( " $\"%28r%5E8-1%29%2F%28r%5E4-1%29=17\"$
\n" ); document.write( " $\"r%5E4%2B1=17\"$
\n" ); document.write( " $\"r%5E4=17-1\"$
\n" ); document.write( " $\"r%5E4=16\"$
\n" ); document.write( " $\"r=root%284%2C16%29\"$
\n" ); document.write( " $\"highlight%28r=2%29\"$
\n" ); document.write( "Then $\"b%5B9%5D=b%5B7%5D%2Ar%5E2=960%2A2%5E2=960%2A4=3840\"$
\n" ); document.write( "So, $\"b%5B9%5D=highlight%283840%29\"$ . \n" ); document.write( "

\n" );